Precalculus 4: Exponentials and logarithms

Define the number e and its exponential function using limits and Taylor series. Explore the connections to the binomial theorem, Pascal's triangle, and the irrational value near 2.7.

Explore sigma notation and summation notation, including index, lower and upper limits, and converting sums to sigmas. Apply addition properties—commutativity, associativity, and distributive law—and index shifting.

Explore how Pascal's triangle and the binomial theorem compute natural powers of a plus b, reveal coefficients through rows, and apply distributive law and foil.

Apply the binomial theorem to powers of a difference by using a and minus b with Pascal's triangle, revealing how signs alternate with the exponent's even or odd nature.

Practice raising binomials to positive natural powers using Pascal's triangle, expanding expressions with binomial coefficients and sign patterns. Apply power rules and compare methods as you work through the exercises.

Explore the binomial theorem by linking Pascal's triangle to binomial coefficients n choose k, and applying the formula to expand (a+b)^n, with small-n visuals and symmetry.

Show how Pascal's triangle with numbers and the binomial coefficients describe the same coefficients in the expansion of (a+b)^n, proving their equivalence and symmetry.

Demonstrates a formal induction proof of the binomial theorem, detailing base case, induction step, and how binomial coefficients relate to Pascal's triangle.

Explore a combinatorial proof of the binomial theorem and the roles of permutations, variations, and combinations. Learn how n choose k and factorials underpin binomial coefficients and Pascal's triangle.

Showcases how the binomial theorem proves that the sum of binomial coefficients equals two to the n, and develops a combinatorial, induction-based counting of all n-element subsets using binary coding.

Explain the axiom of completeness via supremum and infimum, describing upper and lower bounds and the existence of a real number between these sets, with the sqrt(2) example.

Advance the convergence of monotone sequences: prove that increasing and bounded above sequences converge to their supremum and decreasing sequences to their infimum, with an epsilon-based sketch of the proof.

Study the convergence of geometric sequences and series with quotient q in (0,1), using the squeeze theorem to derive the geometric sum and its role toward defining e.

Explore the advanced motivation of the definition and irrationality of e, proving A_n and B_n increase and are bounded by 3, with limits equal to e, using the binomial theorem.

Outline the plan to learn exponential, power, and logarithmic functions, building from basics to plotting f(x)=2^x and comparing with 2^x and 3^x curves.

Master two product rules for powers: (ab)^n = a^n b^n and a^n a^m = a^{n+m}, illustrated with n=3, m=4 and same-base cases to show how exponents add.

Learn two quotient rules, derived from product rules, for (a/b)^n and a^n/a^m with nonzero bases and positive integer exponents, plus practical examples.

Master the power rule by multiplying exponents and applying it to powers of powers, products, and sums, with binomial theorem examples of x and a.

Explore how to extend powers to non-natural exponents while preserving product, quotient, and power rules, with steps toward zero, negative, rational, and real exponents.

Extend the definition of a^x to all real x by applying exponent rules at x = 0, so a^0 = 1 for nonzero bases, and distinguish undefined from indeterminate forms.

Define a to the minus m as the reciprocal of a to the m for nonzero a, yielding negative integer exponents and symmetry with positive powers along the y-axis.

Define rational powers using the nth root, with a^(1/n) as the positive root (a≥0 for even n) and a^(m/n) via the power rule.

Investigate the existence of nth roots by linking continuous power functions to their inverses, guided by the axiom of completeness and supported by a formal article.

Study powers with rational exponents by defining a^x for positive a and rational x, demonstrating monotonic behavior: increasing if a>1, decreasing if 0<a<1.

Explore powers with real exponents and the graphs of exponential functions, defining a^x for irrational x via rational bounds, and show monotonicity for a>1 and decreasing for 0<a<1.

Express numbers as powers with integer bases by factoring, applying the product rule, and combining exponents, yielding 3^(9/2) and 3^(5/4) in this session.

Express numbers as powers with integer bases using product, quotient, and power rules; simplify problems with bases five and three to finalize results like 5^1 and 3^-1.

Compute powers with rational exponents to express a number as a power with integer base, using product and power rules to simplify roots to five to the three-fourths.

Compute various powers by factoring cube roots to simplify the expression, showing the result reduces to zero, a rational number.

Simplify expressions with rational exponents using the power rule and the quotient rule for x>0. Conclude with x^(19/12) or the 12th root of x^19.

Learn to simplify expressions with non-negative a, b, and c by applying rational powers and the power and product rules to combine exponents, yielding ab^2 sqrt(BC).

Simplify the expression by turning square roots into powers and using quotient and power rules to express the left-hand side as a power of x/y, yielding p = 3/8.

Analyze simplifying expressions using exponent rules for positive real numbers and roots. Determine which equalities hold for all a>0, applying product, quotient, and power rules.

Apply the difference-of-squares formula with x = sqrt(ab) and y = sqrt(bc); cancel terms to show the result is not equal to a, b, or c.

Simplify the expression 4^x / 8^y by rewriting with base two and applying power and quotient rules. Since 2x − 3y = 4, the expression evaluates to 2^4 = 16.

Solve careful with signs in problem 14: when a is negative, sqrt(a^2) equals |a|, so the nested roots become the fourth root of -a, confirming option three.

Compute f(0) = sqrt(a^2) − sqrt((-a)^2) and note sqrt(a^2) = |a|, sqrt((-a)^2) = |a|, hence f(0) = 0.

Learn to compare two powers by using the power rule to rewrite 2^2000 as 32^400 and 10^800 as 100^400, proving 2^2000 < 10^800.

Compare two expressions using power and product rules for real exponents, including square roots and difference of squares, to determine which is larger without a calculator.

Explore comparing numbers using exponential bases between 0 and 1 and greater than 1, show inequalities are false by equality of bases via conjugates, and introduce power function reasoning.

Multiply each fraction by the conjugate of its denominator. Use the difference of squares to obtain a common three in the denominators and prove the sum is an integer.

Apply the conjugate method to simplify expressions with square and fourth roots by multiplying by the conjugate, using the difference of squares, canceling terms, and obtaining a simple result.

Demonstrate a three-step method for sums of square roots in the charms of squares and conjugates, verifying sign, squaring, and deducing m; problem 22 gives m equals 3.

Explore problem 23 in the charms of squares and conjugates. Rewrite with square roots, apply the difference of squares to m^2, and conclude m = -2.

Demonstrate how squares and conjugates simplify a problem with square roots (problem 24) to establish that m is negative and m = -2 by computing m^2.

Apply squares and conjugates to problem 25 by determining m from m^2 using the difference of squares and choosing the positive root, noting results may be non-integers.

Explore how precalculus uses squares and conjugates to simplify a tricky expression to 1 plus the square root of 2, using square roots, half exponents, and the product of conjugates.

Use m equals a plus b and Pascal's triangle to compute m^3, derive the polynomial x^3 plus 3abx minus (a^3 plus b^3) equals zero, and conclude m equals one.

Use the sum and difference of cube roots to form a cubic, determine ab=1 and a^3+b^3=18, solve for m, and confirm the unique real root m=3.

Apply the difference of two cubes to a=cube root of five and b=cube root of four to find 1/(a-b). Relate this to removing radicals from the denominator using conjugates.

Interpret towers of exponents using the power rule and recognize that 2^t is strictly increasing, so equal powers have equal exponents in exponential equations, giving x = 2^12.

Determine the domain of the fractional expression raised to the one-half, showing a must be greater than minus five; then simplify the expression to a constant value of two.

Explore power functions from monomials to fractional exponents, and see how the product of power functions remains a power function while sums may not.

Explore power functions with positive natural exponents, focusing on monic monomials, their continuity, parity, monotonicity, domain, range, inverses, and graph scaling.

Explore power functions with negative integer exponents, y = x^-n: reciprocal of x^n, two-branch hyperbolas with axis asymptotes, domain excludes zero, and parity-driven symmetry with monotonicity patterns.

Examine power functions with positive and negative fractional exponents, noting domain restrictions and vertical asymptotes for negative exponents. Explore monotonicity, inverses, and cusps with examples like cube roots and square roots.

Explore power functions with real exponents in precalculus, examining bases between 0 and 1 and greater than 1 and graphs like x^2, x^3, x^e, and x^pi.

Determine a power function from two points by solving for alpha and c, then verify the result by substitution to ensure both points lie on the graph.

Determine c and alpha in power function y = c x^alpha from two points; get c = -2 and alpha = -2, so y = -2 x^{-2}, verified by points.

Determine the power function from two points (4,-6) and (9,-9) by solving for c and alpha; alpha = 1/2 and c = -3, giving f(x) = -3 sqrt(x).

Explore typical graph transformations of power functions, including shifts, reflections, and scaling, illustrated through step-by-step examples from x^3 to 1/x and sqrt(x).

Explore the properties and graphs of exponential functions, including bases greater than one and between zero and one, and their monotonicity, symmetry, and the y-intercept at (0,1).

Analyze comparing powers with the same base via exponential behavior: for base five, the function 5^x is increasing, so larger exponents give larger values, with pi as the greatest.

Compare powers with the same base less than one, noting smaller exponents yield larger values. The exponents sqrt2, 1+sqrt3, pi, 3sqrt2 correspond to powers from least to greatest.

Solve an exponential inequality with base a>1 by using the power rule and a^-1; since a^x is increasing, the inequality becomes x^3 + x^2 - 1 > 0.

Explore properties of exponential functions with positive base a, showing that when the inequalities have the same solution sets, f(x)=a^x is decreasing, so 0<a<1, and the answer is three.

In this problem, identify f(x)=c a^x from the graph, find c=1 and a=1/3, then compute g(-1/2)=f(1/2)=(1/3)^{1/2}=√3/3.

Explore exponential functions and a clever trick for problem 8, linking f and g through 25 = 5^2 and the square rule to find g(x0) = 6.

Explore exponential function graphs and their transformations, including shifting and scaling the input, vertical shifts, and reflections for bases under one, with attention to domain, range, and horizontal asymptotes.

Explore how exponential functions compose with polynomials by rewriting f(x) as p(g(x)) with g(x)=3^x and p(t)=-t^2-2t+8. For t>0, the range of f is (-∞, 8).

Explore the range of a composite function f(x) = p(7^x), where p(t) = t^2 - t - 6 and t > 0, showing the range is [-25/4, infinity).

Determine the range of f(x) = (1/2)^{p(x)} on [1, 4.5], with p(x) = x^2 - 4x, by using t_min = -4 and t_max = 9/4.

Plot a piecewise exponential function formed by expressions with absolute values, identify turning points at 0 and 1, and describe constant and decreasing segments.

Explore an exponential function with base one half and a piecewise exponent involving absolute values. Identify turning points at -2 and 0 and describe a three-part graph: increasing, constant, decreasing.

Explore exponential functions through problem 16 by graphing y = |-(2^{x-5}) + 1|: shift the base graph five units right, reflect across the x-axis, shift up, then apply the absolute value.

Use graphical methods to solve 2^x = cos x by finding intersection points of the exponential and cosine curves, revealing infinitely many solutions and no positive solutions.

Explore how exponential graphs with different bases relate, showing that larger bases dominate positive exponents, smaller bases dominate negative exponents, and all curves meet at x=0, y=1.

Explore the interdependencies between power and exponential functions, focusing on monotonicity and how base and exponent roles switch between the two.

Explore strictly monotone functions, showing how increasing or decreasing behavior leads to an if and only if characterization and practical intuition for monotonicity in precalculus.

Proves that the inverse of a strictly monotone function is strictly monotone and that such functions are invertible via injectivity. Illustrates with square and square-root inverses and domain considerations.

Learn that the reciprocal of a strictly monotone, sign-constant function is monotone in the opposite direction. The lecture illustrates this with e^x, e^{-x}, and x^{2/3}, and includes a brief proof.

Explore how scaling a strictly monotone function by a positive scalar preserves its monotonicity, while multiplying by a negative scalar reverses it, with illustrative graphs and precalculus background on inequalities.

Explore how composing strictly monotone functions affects monotonicity: same-type pairs (increasing with increasing or decreasing with decreasing) yield increasing outputs; opposite-type pairs yield decreasing outputs, with domain considerations.

Explore sums of strictly monotone functions, proving that the sum of two increasing or two decreasing functions remains monotone, and examine domain and mixed monotonicity cases.

Show that the product of strictly monotone positive functions is monotone when both are increasing or both are decreasing, using a mixed product trick on the intersection of their domains.

Explore how strictly monotone functions yield at most one solution to f(x) = d. An increasing or decreasing function is injective, so the solution set is one value or empty.

Explore solving an optional equation with strictly monotone functions by determining the domain, testing a candidate solution, and proving the solution is unique via a strictly increasing left-hand side.

solve equations with strictly monotone exponential functions by dividing by 2^x to obtain constant terms, analyze bases between 0 and 1, and verify a unique solution x=2.

Identify the domain as non-negative x and use the increasing left-hand side, composed of sqrt, |x|, x^2, and 2x, to obtain the unique solution x = 9 with value 111.

Solve with strictly monotone functions by intersecting domains; show the left-hand side is increasing, yielding a unique solution at x = 1, the eighth root of two.

Introduce logarithms as the inverse of exponentials, showing how they turn products into sums and quotients into differences, and highlight base concepts plus real-world applications like pH and decibels.

Explore inverse operations to taking powers by contrasting power and exponential functions, examining when inverses exist, and understanding domain restrictions that affect invertibility, with a preview of logarithmic functions.

Explore logarithms as the inverse of exponentiation, learn base notation (including log, ln, and base ten), and master the cancelling exponentials identities a^(log_a b)=b and log_a(a^b)=b.

Explore logarithms through exercises that reinforce the definition and the link to exponentials. Practice with bases such as 10, e, and 3, using the power rule and book answers.

Define logarithmic functions as the inverse of exponential functions and analyze their properties, including bases positive and not equal to one, monotonicity, and domain and range relationships.

Explore three cool properties of logarithms with base a and its reciprocal, illustrated graphically and proven with a method based on exponential monotonicity.

Explore how logarithmic curves relate to exponential graphs via symmetry along y=x. For bases greater than one, bases yield higher curves for positive arguments, with order reversing through the origin.

Determine whether a logarithm is positive, negative, or zero by comparing the base and the argument, using x = log_a(c) and the base’s relation to one.

Master the product rule for logarithms: log_a(xy) = log_a(x) + log_a(y) for a > 0, a ≠ 1, derived from exponent rules and the exponential 1-to-1 property.

Explore the quotient rule for logarithms with base a, proving log_a(x/y) = log_a x - log_a y using exponential monotonicity and power rules.

Master the power rule for logarithms, derive it from exponent rules, and apply product and quotient rules through plenty of guided exercises and a historical note.

Explains the base change rule for logarithms and powers, derives product and quotient forms, and shows how to express any exponential with base e for problem solving.

Learn the base switch rule for logarithms, a base change special case, showing log_a(b) = 1/log_b(a) for positive bases not equal to one. Apply to four problems on page 446.

Explores typical graph transformations of logarithmic functions by using auxiliary graphs and applying shifts, scaling, and reflections, including base changes and the symmetry with exponentials.

Explore logarithmic functions through domain analysis and inverses, then compare log x^2 and 2 log x graphs, with guided problem-solving practice.

Explore logarithms through properties and graphs, showing log base m of n is irrational for relatively prime m and n, with an application to log base three of two.

Use logarithm definitions to express x, y, z as powers of two, apply the cubic root of their product, and use power rules to obtain the geometric mean 32.

Show that m times n, defined via logarithms, equals the square of a natural number by computing m and n and verifying the product.

Express log 108, log 5, and log base 2 of 20 in terms of a and b, where a=log 3 and b=log 2, using product, quotient, and power rules.

Apply change of base to log_3 10 in terms of a = log_6 2 and b = log_6 5, giving log_3 10 = (a + 2b)/(1 − a).

Analyze logarithms in base six for problem seven, using quotient rule and change of base, then apply monotonicity and the graph of log base six to determine true statements.

Solve exponential equations with the natural logarithm, using ln's 1-to-1 property to equate x with log of the right-hand side, and verify equivalence via rationalization and the difference of squares.

Apply the product rule to logs: log a + log b + log c = log(a b c). Conclude a b c must equal 1 when the sum equals zero.

Use the base switch to turn 1/a, 1/b, 1/c into log_10 3, log_10 5, log_10 15; their sum equals log_10 225, which sits between 2 and 3.

Use base change and base switch formulas to simplify the product of logarithms, showing the result is 3 and is rational, an integer, and less than 5.

Apply the base change formula for logarithms to rewrite x^{log_y z} as z, then use the power rule and cancellation to show the expression equals zero.

Evaluate which logarithmic formulas hold for all x not equal to zero and show that log(1/|x|) equals minus log|x| using the power rule with alpha minus one.

Examine the domain of f(x)=1/log(log(x^2-1)) as a reciprocal of a double logarithmic composition, enforcing inner and outer natural-logarithm domains and excluding division by zero and x^2 = e+1.

Determine the domain of f(x) = ln(2/3 - 3^x) from 2/3 - 3^x > 0. Conclude the largest integer x with 3^x < 2/3 is -1.

This problem shows that the graphs of f(x)=log(x-1) and g(x)=log(1-x) are symmetric about the vertical line x=1, with x=1 as their common asymptote, and analyzes their domains x>1 and x<1.

Explore the function f(x) = ln(x+√(x^2+1)) defined for all real x; prove it is increasing on R, odd, with f(0)=0, and compare its growth to ln(2x) for large x.

Show that f(x) equals x log((x+3)/(x-3)) is even by using a symmetric domain about zero, noting zeros at ±3, and applying log rules with minus one.

Use the generalized product rule and base switch to turn a sum of logs with base x into log base x of n!, which equals log base n! of x.

Apply the base switching formula to rewrite each exponent as a logarithm base x, then use the generalized product rule to sum the exponents, yielding the constant result 2^45.

Define exponential equations and inequalities as those with the unknown in the exponents, and learn solving methods using examples like 3^x = 7 and varying bases.

Learn that strict monotonicity makes exponential functions 1-to-1, so equal bases imply equal exponents, with inequality directions depending on the base. Use logarithms to solve, via cancellation and inverse properties.

We review solved exponential equations and the interactions between power and exponential functions, using monotonicity and one-to-one properties, and apply a substitution trick to reduce to quadratics.

Explore solving exponential inequalities by rewriting to equal bases, showing that when a>1 the function a^x is increasing and when 0<a<1 it is decreasing, with solutions matching polynomial inequalities.

Solve equations by applying inverse operations in reverse order, from linear forms to exponential ones, using addition, subtraction, multiplication, division, and logarithms.

Explain the meaning of logarithm, its link to ratios and proportions, and show that logarithms of proportional numbers have a constant distance across bases two, base ten, and natural logarithm.

Learn to recognize and solve basic exponential equations by using equivalent equations, logarithms, and the power rule, including equations with different bases and substitution, and practice the book's problems.

Learn to solve exponential equations and inequalities by classifying methods, using properties of exponential functions, power rules, and equal-base reasoning, with simple problems and graphical insight.

Solve an exponential equation with different bases by applying natural logarithms, excluding x=0, to obtain the exact solution (ln3 - 2 ln2)/(ln2 - ln3).

Apply substitution t = 3^x to convert the exponential equation into t^2 -5t +6 = 0, solve t = 2 or 3, then x = log_3(2) or x = 1.

Substitute t = e^{3x} to convert e^{6x}-e^{3x}-6=0 into t^2-t-6=0 with t>0; solve t=3, giving x=(1/3)ln3 as the unique root between 0 and 1.

Solve the exponential equation 9^x - 6^x - 2^{2x+1} = 0 by rewriting with a common exponent, using t = (3/2)^x, solving t^2 - t - 2 = 0, and finding x = ln 2 / ln(3/2).

Solve exponential equations by substitution to convert a rational third-degree polynomial, using t = 2^x, and factor to reveal a unique solution x = 0.

Explore solving an exponential equation by rewriting with regular fractions and powers of two, then equate exponents to find the unique solution x = 8/7.

Learn to solve exponential equations with reciprocal bases using substitution to obtain a quadratic; generalize to a^x + a^{-x} = c with zero, one, or two real solutions.

Learn how to find inverse functions of exponential functions in precalculus by solving equations with logarithms, using bases 3 and 2, and relate inverses to domain and range.

Analyze inverse functions of exponential forms, solve problem 13 in precalculus, and examine the domains of f and f inverse, with composition and logarithmic methods.

Find the inverse of the special function f, defined as (e^x - e^{-x})/2, by substitution; derive f^{-1}(x) = ln(x + sqrt(x^2 + 1)).

Explore graphical and analytical methods to solve functional inequalities, including comparing graphs, identifying intersection points, and using monotonicity of exponential and logarithmic functions.

Solve problem 15: inequality 0.5^(x^2−3) < (1/4)^x by rewriting to base 1/2 and using its decreasing nature, yielding x < -1 or x > 3.

This lecture solves two exponential inequalities (problem 16) using monotonicity and substitution; it then analyzes a second inequality with base 1/3 via t=(1/3)^x to obtain x<-2 or x>0.

This lecture solves a tough exponential inequality by aligning bases to three and using a substitution t = fourth root of x, yielding 0 ≤ x < 16.

Explore logarithmic equations and inequalities by analyzing domains, bases, and arguments; identify extraneous roots and apply monotonicity to solve for bases greater than one or between zero and one.

Discover how to find inverse functions for functions defined by logarithms by solving equations, with domain considerations, and graph verification using base 4 and base 10 examples.

Solve a logarithmic equation using natural logs and the quotient rule, determine the domain x>1, and obtain x = 1 + sqrt(2) as the valid solution.

Apply the power and product rules to solve a logarithmic equation, use t = e^x for a quadratic, enforce x>0, and verify that x = ln 5 satisfies the equation.

Explore solving logarithmic equations with domain analysis and warnings about log rules for sums or quotients, noting squaring may introduce extraneous solutions; the valid root is x=5.

Set domain restrictions, apply the base-switch and power rules, substitute t = log_4 x to obtain a quadratic, solve for t, and derive x values: 4^(-1/3) and 8.

Explore logarithmic inequalities with base two, determine the domain from x^2−x>0, apply the quotient rule to reduce to a quadratic, and obtain the solution intervals (-1,0) and (1,2).

Solve logarithmic inequalities by analyzing the domain and substituting t = log_3 x; deduce x in (0,3) or x ≥ 27, excluding x = 3.

Solves a logarithmic inequality with a reciprocal base, uses change of base and power rule, analyzes cases x in (0,1) and x>1, deduces domain and final solution (0,1) ∪ (2,∞).

Solve logarithmic inequalities with different bases by applying the change-of-base formula to base three and simplifying. Determine the domain x>0 and conclude x<27, so the solution is (0,27).

Explore applications of exponential and logarithmic functions through focused book reading and concise videos. Learn to solve exponential and logarithmic equations and apply concepts in science, engineering, and finance.

Explore growth and decay through percent changes and change factors, then apply multiplier concepts to price, area, and volume changes for practical calculations.

Explore geometric series and annuity in precalculus, deriving the sum of first n+1 terms of a geometric sequence and applying to simple and compound interest.

Explore compound interest within exponential functions using the change factor, principal, and rate. See how compounding frequency from yearly to continuous affects the amount after t years and doubling time.

Explore how e arises from continuously compounded interest, via limits and a lemma about the indeterminate form (1+ a_n)^{1/a_n} tending to e, connecting to the exponential function.

Explore how an exponential decay models a car’s value with V(x)=25(4/5)^x, starting at 25k and approaching zero. Rewrite with base e to reveal a yearly decay of about 22.3%.

Explore how coffee cools over time using the Newton's law of cooling model T(t) = 70 + 90 e^{-0.1 t}, with a horizontal asymptote at 70 Fahrenheit.

Examine radioactive decay through the exponential model a(t)=a0 e^{kt}, with k negative, and analyze half-life, deriving h from k (and vice versa) using the book’s examples.

Learn about uninhibited exponential growth using the model n(t) = n0 e^{kt}, determine doubling time, and apply to world population and cell growth examples.

See how the linear and logarithmic scales reveal different relations: equal distances on a linear scale reflect differences, while on a logarithmic scale they reflect quotients for exponential graphs.

Explore how logarithms determine earthquake magnitude by linking seismograph readings to the Richter scale, and learn to solve equations and inequalities for seismograph measurements.

Explore logarithmic functions in the decibel scale to measure sound intensity. Solve logarithmic equations and a single inequality to find the intensity corresponding to a given decibel level.

Investigate logarithmic functions and measuring acidity by examining pH and hydrogen concentration, with a solved pH example and guidance on exponential and logarithmic methods.