Properties of Logarithms

We've explored the product rule and the quotient rule, but how can we take the logarithm of a power, such as \(x^2\)? One method is as follows:

\(\begin{aligned}

\log _{b}\left(x^{2}\right) &=\log _{b}(x \cdot x) \\

&=\log _{b} x+\log _{b} x \\

&=2 \log _{b} x

\end{aligned}\)

Notice that we used the product rule for logarithms to find a solution for the example above. By doing so, we have derived the power rule for logarithms, which says that the log of a power is equal to the exponent times the log of the base. Keep in mind that, although the input to a logarithm may not be written as a power, we may be able to change it to a power. For example,

\(100=10^2 \quad \sqrt{3}=3 \frac{1}{2} \quad \frac{1}{e}=e^{−1}\)

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.