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Precalculus Unit 5: Exponential & logarithmic functions

Exponential functions change by equal factors over equal intervals. Logarithms invert them and turn multiplication into addition.

Covers: Sequences vs functions, exponential models, log properties, semi-log plots

Study guide

Sequences connect to functions. An arithmetic sequence (constant difference) is a linear function restricted to whole-number inputs, and a geometric sequence (constant ratio) is an exponential function restricted the same way. Over equal input steps, linear functions change by equal amounts and exponential functions change by equal factors.

Exponential models have the form f(x) = a·bˣ, with initial value a and base b. If b > 1 the function grows, and the growth factor is 1 + r for a percent rate r; if 0 < b < 1 it decays. Exponential functions have a horizontal asymptote (y = 0 for the basic form) and grow faster than any polynomial in the long run. The natural base e appears in continuous growth, f(t) = ae^(kt).

Logarithms are the inverses of exponentials: log_b x = y means bʸ = x. The natural log ln x has base e, so ln is the inverse of eˣ. Log properties turn products into sums, quotients into differences, and powers into multiples: log(xy) = log x + log y, log(x/y) = log x − log y, log(xⁿ) = n log x. Use them to solve exponential equations: take the log of both sides and bring the exponent down.

On a semi-log plot, the y-axis is logarithmic. Exponential data appear as a straight line there, and the slope of that line relates to the growth rate, which is a quick test for exponential behavior.

To find a and b in f(x) = a·bˣ from two points, substitute both points to get two equations, then divide one equation by the other to cancel a and solve for b. Substitute b back into either equation to find a. Exponent rules let you rewrite exponential expressions: b^(x+k) = bᵏ·bˣ, so a horizontal shift of an exponential is the same as a vertical dilation, and b^(x/k) = (b^(1/k))ˣ changes the growth factor per unit. Use these to put an expression in the form a·bˣ.

An equation like e^(2x) − 5eˣ + 6 = 0 is quadratic in eˣ: let u = eˣ, solve u² − 5u + 6 = 0, then solve eˣ = u for each root. Reject any u ≤ 0, because eˣ is always positive. After condensing logs and converting to exponential form, check each solution in the original equation. Any value that makes a log's input zero or negative is extraneous.

An arithmetic series of n terms sums to Sₙ = n(a₁ + aₙ)/2, and a finite geometric series sums to Sₙ = a₁(1 − rⁿ)/(1 − r). An infinite geometric series with |r| < 1 sums to a₁/(1 − r), which is used in problems such as the total distance of a bouncing ball.

Key ideas

• Arithmetic sequences grow linearly; geometric sequences grow exponentially.
• f(x) = a·bˣ: a is the initial value, b is the growth factor.
• Growth rate r and growth factor b are related by b = 1 + r.
• Logarithms answer "what exponent gives this value?"
• On a semi-log plot (log y vs x), exponential data look linear.

Formulas and rules

Common mistakes

• Mixing up the growth rate (0.05) with the growth factor (1.05).
• Forgetting log properties don't apply to sums inside the log.

Words to know

Growth factor

1 + r (8% growth → × 1.08)

Semi-log plot

log(y) vs. x; exponential data look linear.

Natural logarithm (ln x)

The inverse of eˣ; log base e.

Equal bases property

If bˣ = bʸ, then x = y (2ˣ = 32 = 2⁵ → x = 5).

Constant ratio property

Exponential outputs over equal x-steps share a constant ratio.

Arithmetic sequence

Constant differences; a linear function on whole numbers.

Geometric sequence

Constant ratios; an exponential function on whole numbers.

Exponential model

f(x) = a·bˣ

Logarithm

log_b x = y means bʸ = x.

Change of base

log_b x = ln x/ln b

Continuous growth

f(t) = ae^(kt)

Sample practice questions

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