Convert from exponential to logarithmic form

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.

  1. [latex]{2}^{3}=8\\[/latex]
  2. [latex]{5}^{2}=25\\[/latex]
  3. [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].

  1. [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].

  2. [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].

  3. [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]

Solution