2.13 Exponential and Logarithmic Equations and Inequalities

Solving exponential equations

To solve exponential equations, isolate the exponential expression, then take the logarithm of both sides (or rewrite both sides with the same base). If b^x = c, then x = log_b(c). Since exponential functions are one-to-one, this always gives a single valid solution when c > 0. Check the domain: exponential outputs must be positive.

Solving logarithmic equations

To solve logarithmic equations, combine log terms using properties (product, quotient, power), then rewrite in exponential form to eliminate the log. Always check for extraneous solutions — logs are only defined for positive arguments, so any solution making an argument ≤ 0 must be rejected.

Inequalities and growth comparisons

For exponential and logarithmic inequalities, use the fact that exponential functions are strictly increasing (if b > 1) to preserve inequality direction when taking logs of both sides. These equations model real contexts like finding when a population exceeds a value or when an investment surpasses a target, often requiring a calculator for non-nice bases.

Key terms

How do you solve b^x = c for x?

Take log_b of both sides: x = log_b(c), or rewrite both sides with a common base.

Why must you check solutions to logarithmic equations?

Because logarithms are only defined for positive arguments — extraneous solutions can arise that make the argument ≤ 0.

Product property of logs

log_b(mn) = log_b(m) + log_b(n).

Solving an exponential inequality b^x > c (b > 1)

Take log of both sides; since the function is increasing, the inequality direction stays the same: x > log_b(c).

Why are exponential/logarithmic equations useful in modeling?

They let us solve for time, rate, or amount in real-world growth/decay contexts, like population growth or investment value.

Try an AP-style question

Solve 3^(x + 1) = 27.

1. x = 2
2. x = 9
3. x = 26
4. x = 3