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Algebra 2 Unit 6: Exponential & logarithmic functions

Exponential functions model percent growth; logarithms answer "what exponent?" Together they solve problems like doubling time.

Covers: Growth and decay models, e and continuous growth, logarithms and their properties, solving exponential and log equations

Study guide

Exponential functions y = a·bˣ model growth (b > 1) and decay (0 < b < 1). With a percent rate r, growth is a(1 + r)ᵗ and decay is a(1 − r)ᵗ. Compound interest is A = P(1 + r/n)^(nt) for n compounding periods per year. As n grows, this approaches continuous compounding, A = Pe^(rt), where e ≈ 2.718 is the natural base. Half-life and doubling-time problems fit the same models.

A logarithm is an exponent: log_b x = y means bʸ = x. The common log, log x, uses base 10, and the natural log, ln x, uses base e. Logarithmic functions are inverses of exponential functions, so their graphs reflect across y = x, and the domain of log x is x > 0.

The log properties mirror exponent rules: the product rule log(xy) = log x + log y, the quotient rule log(x/y) = log x − log y, and the power rule log(xⁿ) = n·log x. The change-of-base formula, log_b x = ln x / ln b, lets you evaluate any log on a calculator.

To solve an exponential equation, rewrite both sides with the same base if possible, or take a log of both sides and use the power rule to bring the exponent down. To solve a log equation, condense the logs into one and rewrite in exponential form. Check that log equation answers don't make any log's input zero or negative.

The graph of f(x) = a·bˣ + k has the horizontal asymptote y = k, its y-intercept is a + k, and if a > 0 its range is y > k. Shifting left or right changes neither the asymptote nor the range. To change a growth factor per period to a factor per year, use exponent rules: if a quantity grows by a factor of B every 5 years, the annual factor is B^(1/5), and the annual rate is B^(1/5) − 1. Similarly, an annual factor b becomes a monthly factor b^(1/12).

A continuous model Ae^(kt) can be rewritten as A(eᵏ)ᵗ, so its growth factor per unit of time is b = eᵏ and its percent rate per unit of time is eᵏ − 1. Going the other way, b = eᵏ means k = ln b. If a substance has half-life h, the amount after time t is A = A₀(1/2)^(t/h). Doubling time works the same way with base 2: A = A₀·2^(t/d).

To fit an exponential model to data, enter the data and run exponential regression to get y = a·bˣ, rounding as directed. b − 1 is the growth rate if b > 1, and 1 − b is the decay rate if b < 1.

Key ideas

• log_b x = y means bʸ = x.
• Log properties: product → sum, quotient → difference, power → coefficient.
• Change of base: log_b x = ln x / ln b.
• Solve aˣ = c by taking logs of both sides.
• Check that log equation solutions keep every argument positive.

Formulas and rules

Common mistakes

• log(x + y) ≠ log x + log y.
• Keeping a negative solution in a log equation.

Words to know

Common logarithm

log x means log₁₀ x (log 1000 = 3).

log_b(xⁿ)

n·log_b x

Change-of-base formula

log_b x = ln x / ln b

Continuous compounding formula

A = Pe^(rt)

Domain of log x

x > 0

Natural base e

e ≈ 2.718, the base of continuous growth.

Logarithm

log_b x = y means bʸ = x.

Natural logarithm

ln x = logₑ x.

Product rule for logs

log(xy) = log x + log y

Quotient rule for logs

log(x/y) = log x − log y

Compound interest

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

Half-life model

A = A₀(1/2)^(t/h)

Solving exponential equations

Use a common base, or take a log of both sides and bring the exponent down.

Solving log equations

Condense to one log, rewrite in exponential form, check the answers.

Sample practice questions

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