Contextual Applications of Derivatives: Background You’ll Need 2

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Contextual Applications of Derivatives: Background You’ll Need 2

Use the power rule to simplify logarithmic expressions

Using the Power Rule for Logarithms

The power rule for logarithms is a fundamental concept that simplifies the process of working with logarithmic expressions involving powers.

How can we take the logarithm of a power, such as ${x}^{2}$ ? One method is as follows:

$\begin{array}{l}{\mathrm{log}}_{b}\left({x}^{2}\right)\hfill & ={\mathrm{log}}_{b}\left(x\cdot x\right)\hfill \\ \hfill & ={\mathrm{log}}_{b}x+{\mathrm{log}}_{b}x\hfill \\ \hfill & =2{\mathrm{log}}_{b}x\hfill \end{array}$

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.

$\begin{array}{lll}100={10}^{2}, \hfill & \sqrt{3}={3}^{\frac{1}{2}}, \hfill & \frac{1}{e}={e}^{-1}\hfill \end{array}$

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.

${\mathrm{log}}_{b}\left({M}^{n}\right)=n{\mathrm{log}}_{b}M$

Rewrite ${\mathrm{log}}_{2}{x}^{5}$ .

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.

${\mathrm{log}}_{2}\left({x}^{5}\right)=5{\mathrm{log}}_{2}x$

Rewrite ${\mathrm{log}}_{3}\left(25\right)$ using the power rule for logs.

Expressing the argument as a power, we get ${\mathrm{log}}_{3}\left(25\right)={\mathrm{log}}_{3}\left({5}^{2}\right)$ .

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.

${\mathrm{log}}_{3}\left({5}^{2}\right)=2{\mathrm{log}}_{3}\left(5\right)$