2.12 Logarithmic Function Manipulation

Properties of Logarithms

Because logarithms are inverses of exponentials, they turn products into sums and powers into products: log(ab) = log(a) + log(b), log(a/b) = log(a) − log(b), and log(a^k) = k·log(a). These properties let you rewrite complicated logarithmic expressions as simpler sums, differences, or multiples of simpler logs — a key skill for solving equations and analyzing rates of change.

Combining and Expanding Logs

Expanding breaks one log of a product/quotient/power into multiple simpler logs; condensing (combining) reverses this, merging sums/differences of logs into one log using the same properties. Condensing is especially useful for solving equations like log(x) + log(x−3) = 1, since you can rewrite the left side as a single log and then exponentiate both sides.

Change of Base

Since calculators typically only compute base-10 or base-e logs, the change of base formula lets you evaluate any log: log_b(x) = log(x)/log(b) = ln(x)/ln(b). This is essential for evaluating logs with unusual bases and for comparing growth rates of exponential functions with different bases.

Key terms

Product Property of Logs

log(ab) = log(a) + log(b)

Quotient Property of Logs

log(a/b) = log(a) − log(b)

Power Property of Logs

log(a^k) = k·log(a)

Change of Base Formula

log_b(x) = ln(x)/ln(b) = log(x)/log(b)

Condensing logs

Combining sums/differences of logs into a single log using log properties, often to solve equations.

Try an AP-style question

log 2 + log 5 = ?

1. log 2.5
2. 10
3. log 7
4. 1