Using the Power Rule for Logarithms
We have explored the product rule and the quotient rule, but how can we take the logarithm of a power, such as [latex]{x}^{2}[/latex]? One method is as follows:
[latex]\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}[/latex]
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.
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.
[latex]{\mathrm{log}}_{b}\left({M}^{n}\right)=n{\mathrm{log}}_{b}M[/latex]
- Express the argument as a power, if needed.
- Write the equivalent expression by multiplying the exponent times the logarithm of the base.
- Product Rule [latex]{a}^{m}\cdot {a}^{n}={a}^{m+n}[/latex]
- Quotient Rule [latex]\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}[/latex]
- Power Rule [latex]{\left({a}^{m}\right)}^{n}={a}^{m\cdot n}[/latex]
- Zero Exponent [latex]{a}^{0}=1[/latex]
- Negative Exponent [latex]{a}^{-n}=\dfrac{1}{{a}^{n}} \text{ and } {a}^{n}=\dfrac{1}{{a}^{-n}}[/latex]
- Power of a Product [latex]\large{\left(ab\right)}^{n}={a}^{n}{b}^{n}[/latex]
- Power of a Quotient [latex]\large{\left(\dfrac{a}{b}\right)}^{n}=\dfrac{{a}^{n}}{{b}^{n}}[/latex]