To convert from exponents to logarithms, we follow the same steps in reverse. We identify the base b, exponent x, and output y. Then we write [latex]x={\mathrm{log}}_{b}\left(y\right)[/latex].
Example 2: Converting from Exponential Form to Logarithmic Form
Write the following exponential equations in logarithmic form.
- [latex]{2}^{3}=8[/latex]
- [latex]{5}^{2}=25[/latex]
- [latex]{10}^{-4}=\frac{1}{10,000}[/latex]
Solution
First, identify the values of b, y, and x. Then, write the equation in the form [latex]x={\mathrm{log}}_{b}\left(y\right)[/latex].
- [latex]{2}^{3}=8[/latex]
Here, b = 2, x = 3, and y = 8. Therefore, the equation [latex]{2}^{3}=8[/latex] is equivalent to [latex]{\mathrm{log}}_{2}\left(8\right)=3[/latex].
- [latex]{5}^{2}=25[/latex]
Here, b = 5, x = 2, and y = 25. Therefore, the equation [latex]{5}^{2}=25[/latex] is equivalent to [latex]{\mathrm{log}}_{5}\left(25\right)=2[/latex].
- [latex]{10}^{-4}=\frac{1}{10,000}[/latex]
Here, b = 10, x = –4, and [latex]y=\frac{1}{10,000}[/latex]. Therefore, the equation [latex]{10}^{-4}=\frac{1}{10,000}[/latex] is equivalent to [latex]{\text{log}}_{10}\left(\frac{1}{10,000}\right)=-4[/latex].
Try It 2
Write the following exponential equations in logarithmic form.
a. [latex]{3}^{2}=9[/latex]
b. [latex]{5}^{3}=125[/latex]
c. [latex]{2}^{-1}=\frac{1}{2}[/latex]