Properties of Logarithms

You’ll be able to

Interpret a logarithm as the exponent that produces a given number
Apply the product, quotient, and power rules for logarithms
Evaluate simple logarithms without a calculator

A logarithm is an exponent

The statement log_b(y) = x means exactly bˣ = y: the logarithm answers the question “to what power must I raise the base b to get y?” So log₂(8) asks “2 to what power is 8?” — and since 2³ = 8, the answer is 3. Logarithms and exponentials are two views of the same relationship.

The three power laws, restated

Because logs are exponents, the exponent rules become log laws. The product rule turns a product inside the log into a sum: log(xy) = log x + log y. The quotient rule turns a quotient into a difference: log(x/y) = log x − log y. The power rule brings an exponent out front: log(xⁿ) = n·log x. These let you expand or condense logarithmic expressions.

Logarithm laws

log(xy) = log x + log y · log(x/y) = log x − log y · log(xⁿ) = n·log x

Products become sums, quotients become differences, and exponents come out front as multipliers. Note that log(x + y) does NOT simplify.

Worked example

Evaluate log₂(8) using the definition of a logarithm.

1.Rewrite log₂(8) = x as the equivalent exponential equation 2ˣ = 8.
2.Express 8 as a power of the base 2: 8 = 2³.
3.So 2ˣ = 2³, which forces x = 3.

Answer: log₂(8) = 3, because 2³ = 8. The logarithm returns the exponent needed on the base to reach the argument.

Checkpoint

Which expression is equivalent to log(xy)?

Watch out

The log laws apply to products, quotients, and powers — never to sums. log(x + y) does not equal log x + log y. Only a product inside the log becomes a sum outside it.

Checkpoint

Rewrite log₂(8) as a number.

On the exam

Use the power rule to solve exponential equations: log(bˣ) = x·log b lets you pull the variable out of the exponent. Recognizing when to expand versus condense is what the free-response rubric rewards.

Answer the 2 checkpoints as you read.