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College Algebra Unit 9: Exponential & logarithmic functions

Exponential functions model repeated multiplication, like growth and decay. Logarithms are their inverses and let you solve for an exponent.

Covers: Graphs, log properties, solving exp/log equations, growth & decay

Study guide

Exponential functions f(x) = bˣ (b > 0, b ≠ 1) grow when b > 1 and decay when 0 0. The natural exponential eˣ uses e ≈ 2.718.

Logarithms are inverses of exponentials: log_b x = y means bʸ = x. So log_b x has domain x > 0, passes through (1, 0), and has a vertical asymptote at x = 0. The common log is base 10 and the natural log ln is base e.

Log properties: the product rule log_b(xy) = log_b x + log_b y, the quotient rule log_b(x/y) = log_b x − log_b y, the power rule log_b(xⁿ) = n log_b x, and the change-of-base formula log_b x = ln x/ln b. Use them to expand or condense expressions.

To solve exponential equations, write both sides with the same base and set exponents equal, or take a log of both sides and use the power rule. To solve logarithmic equations, condense to a single log, rewrite in exponential form, and check that no answer makes a log's argument zero or negative.

Growth and decay models include compound interest A = P(1 + r/n)^(nt), continuous compounding A = Pe^(rt), and exponential decay with half-life, A = A₀(1/2)^(t/h).

For y = a·bˣ through (x₁, y₁) and (x₂, y₂), divide the two equations to get b^(x₂ − x₁) = y₂/y₁, so b = (y₂/y₁)^(1/(x₂ − x₁)). Then substitute one point to find a. Exponential growth eventually beats any power function: 2ˣ overtakes x¹⁰ for large enough x, even though x¹⁰ is larger for many smaller values of x.

A quantity that grows r percent per period follows A = A₀(1 + r)ᵗ. One that is multiplied by k every d time units follows A = A₀·k^(t/d), for example tripling every 4 hours.

Key ideas

• f(x) = a·bˣ grows if b > 1 and decays if 0 < b < 1.
• log_b(x) = y means bʸ = x.
• ln is log base e; log with no base usually means base 10.
• Use log properties to expand, condense and solve.
• To solve for an exponent, take a log of both sides; to solve a log equation, rewrite in exponential form and check for extraneous solutions.

Formulas and rules

Common mistakes

• Writing log(x + y) = log x + log y.
• Keeping a solution that makes a log's argument negative.

Words to know

Definition of a logarithm

log_b x = y means bʸ = x.

Product rule for logs

log_b(xy) = log_b x + log_b y

Power rule for logs

log_b(xⁿ) = n·log_b x

Change-of-base formula

log_b x = ln x / ln b

Compound interest formula

A = P(1 + r/n)^(nt)

Domain of ln x

x > 0

Continuous compounding

A = Pe^(rt)

Exponential function

f(x) = bˣ with b > 0, b ≠ 1.

Natural exponential

eˣ, with e ≈ 2.718.

Logarithmic function

log_b x, the inverse of bˣ; domain x > 0.

Quotient rule for logs

log_b(x/y) = log_b x − log_b y

Solving exponential equations

Same base → set exponents equal; otherwise take a log of both sides.

Half-life model

A = A₀(1/2)^(t/h)

Graph of y = bˣ

Passes through (0, 1), asymptote y = 0, rising if b > 1 and falling if 0 < b < 1.

Exponential growth and decay

A = A₀e^(kt): growth if k > 0, decay if k < 0.

Sample practice questions

39 more questions like these, from level 1 to 5, are in the practice set.