Properties of Logarithms

The Power Rule for Logarithms

The power rule for logarithms can be used to simplify the logarithm of a power by rewriting it as the product of the exponent times the logarithm of the base.

$log_b(M^n)=nlog_bM$

HOW TO

Given the logarithm of a power, use the power rule of logarithms to write an equivalent product of a factor and a logarithm.

1. Express the argument as a power, if needed.

2. Write the equivalent expression by multiplying the exponent times the logarithm of the base.

Example 3

Expanding a Logarithm with Powers

Expand $log_2x^5$.

Solution

The argument is already written as a power, so we identify the exponent, 5, and the base, $x$, and rewrite the equivalent expression by multiplying the exponent times the logarithm of the base.

$log_2(x^5)=5log_2x$

Try It #3

Expand $\ln x^2$.

Example 4

Rewriting an Expression as a Power before Using the Power Rule

Expand $log_3(25)$ using the power rule for logs.

Solution

Expressing the argument as a power, we get $log_3(25)=log_3(5^2)$.

Next we identify the exponent, 2, and the base, 5, and rewrite the equivalent expression by multiplying the exponent times the logarithm of the base.

$log_3(5^2)=2log_3(5)$

Try It #4

Expand $\ln (\frac{1}{x^2})$.

Example 5

Using the Power Rule in Reverse

Rewrite $4 \ln (x)$ using the power rule for logs to a single logarithm with a leading coefficient of 1.

Solution

Because the logarithm of a power is the product of the exponent times the logarithm of the base, it follows that the product of a number and a logarithm can be written as a power. For the expression $4 \ln (x)$, we identify the factor, 4, as the exponent and the argument, $x$, as the base, and rewrite the product as a logarithm of a power: $4 \ln (x) = \ln(x^4)$.

Try It #5

Rewrite $2log_34$ using the power rule for logs to a single logarithm with a leading coefficient of 1.