AlgebraAlgebra 2High school

Logarithms

Logarithm laws, change of base and solving exponential equations.

A logarithm answers the question: to what power must the base be raised to give this number? So $\log_2 8 = 3$ because $2^3 = 8$ . Logarithms undo exponentials, which is why they appear whenever the unknown sits in an exponent, as in compound interest, population growth, pH and decibels.

The log rules mirror the exponent rules. Because multiplying powers adds exponents, the log of a product is the sum of the logs. The change-of-base formula $\log_b x = \frac{\ln x}{\ln b}$ lets any calculator evaluate any base.

Key terms

Logarithm: The exponent you raise the base to in order to get the argument.
$\log_b x = y \iff b^y = x$
Change of base: Rewrite any logarithm with a convenient base.
$\log_b x = \frac{\ln x}{\ln b}$

Key ideas

$\log_b(xy) = \log_b x + \log_b y$ and $\log_b \frac{x}{y} = \log_b x - \log_b y$ .
$\log_b(x^k) = k \log_b x$ , the rule that brings an exponent down.
Logs are defined only for positive arguments; check solutions against the domain.
$\ln$ means base $e$ and $\log$ without a base usually means base 10.

Worked example

Solve $3 \cdot 2^x = 48$ .

Divide by 3: $2^x = 16$ .
Take $\log_2$ : $x = \log_2 16$ .
Since $2^4 = 16$ , $x = 4$ .
For non-exact cases, $x = \frac{\ln 16}{\ln 2}$ on a calculator.

Answer: $x = 4$

Common mistakes

Writing $\log(x + y) = \log x + \log y$ .
Taking the log before isolating the exponential term.
Keeping a solution that makes a log argument zero or negative.

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

1.Evaluate $\log_3 81$ .
Show answer
$4$
2.Evaluate $\log_{10} 0.001$ .
Show answer
$-3$
3.Solve $5^x = 125$ .
Show answer
$x = 3$

Keep practising

Calculators

Logarithm Calculator