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Study Guides > College Algebra

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 x=logb(y)x={\mathrm{log}}_{b}\left(y\right).

Example 2: Converting from Exponential Form to Logarithmic Form

Write the following exponential equations in logarithmic form.

  1. 23=8{2}^{3}=8
  2. 52=25{5}^{2}=25
  3. 10−4=110,000{10}^{-4}=\frac{1}{10,000}

Solution

First, identify the values of b, y, and x. Then, write the equation in the form x=logb(y)x={\mathrm{log}}_{b}\left(y\right).

  1. 23=8{2}^{3}=8

    Here, b = 2, x = 3, and y = 8. Therefore, the equation 23=8{2}^{3}=8 is equivalent to log2(8)=3{\mathrm{log}}_{2}\left(8\right)=3.

  2. 52=25{5}^{2}=25

    Here, b = 5, x = 2, and y = 25. Therefore, the equation 52=25{5}^{2}=25 is equivalent to log5(25)=2{\mathrm{log}}_{5}\left(25\right)=2.

  3. 10−4=110,000{10}^{-4}=\frac{1}{10,000}

    Here, b = 10, x = –4, and y=110,000y=\frac{1}{10,000}. Therefore, the equation 10−4=110,000{10}^{-4}=\frac{1}{10,000} is equivalent to log10(110,000)=−4{\text{log}}_{10}\left(\frac{1}{10,000}\right)=-4.

Try It 2

Write the following exponential equations in logarithmic form.

a. 32=9{3}^{2}=9

b. 53=125{5}^{3}=125

c. 2−1=12{2}^{-1}=\frac{1}{2}

Solution

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