Converting Between Logarithmic And Exponential Form
Learning Outcomes
- Convert from logarithmic to exponential form.
- Convert from exponential to logarithmic form.
tip for success
Understanding what a logarithm is requires understanding what an exponent is. A logarithm is an exponent. Read the paragraphs and boxes below carefully, perhaps more than once or twice, to gain the understanding of the inverse relationship between logarithms and exponents. Keep in mind that the inverse of a function effectively "undoes" what the other does. You can use the definition of the logarithm given below to solve certain equations involving exponents and logarithms.A General Note: Definition of the Logarithmic Function
A logarithm base b of a positive number x satisfies the following definition: For [latex]x>0,b>0,b\ne 1[/latex], [latex]y={\mathrm{log}}_{b}\left(x\right)\text{ is equal to }{b}^{y}=x[/latex], where- we read [latex]{\mathrm{log}}_{b}\left(x\right)[/latex] as, "the logarithm with base b of x" or the "log base b of x."
- the logarithm y is the exponent to which b must be raised to get x.
- if no base [latex]b[/latex] is indicated, the base of the logarithm is assumed to be [latex]10[/latex].
- the domain of the logarithm function with base [latex]b \text{ is} \left(0,\infty \right)[/latex].
- the range of the logarithm function with base [latex]b \text{ is} \left(-\infty ,\infty \right)[/latex].
Q & A
Can we take the logarithm of a negative number?
No. Because the base of an exponential function is always positive, no power of that base can ever be negative. We can never take the logarithm of a negative number. Also, we cannot take the logarithm of zero. Calculators may output a log of a negative number when in complex mode, but the log of a negative number is not a real number.How To: Given an equation in logarithmic form [latex]{\mathrm{log}}_{b}\left(x\right)=y[/latex], convert it to exponential form
- Examine the equation [latex]y={\mathrm{log}}_{b}x[/latex] and identify b, y, and x.
- Rewrite [latex]{\mathrm{log}}_{b}x=y[/latex] as [latex]{b}^{y}=x[/latex].
Example: Converting from Logarithmic Form to Exponential Form
Write the following logarithmic equations in exponential form.- [latex]{\mathrm{log}}_{6}\left(\sqrt{6}\right)=\frac{1}{2}[/latex]
- [latex]{\mathrm{log}}_{3}\left(9\right)=2[/latex]
Answer: First, identify the values of b, y, and x. Then, write the equation in the form [latex]{b}^{y}=x[/latex].
- [latex]{\mathrm{log}}_{6}\left(\sqrt{6}\right)=\frac{1}{2}[/latex] Here, [latex]b=6,y=\frac{1}{2},\text{and } x=\sqrt{6}[/latex]. Therefore, the equation [latex]{\mathrm{log}}_{6}\left(\sqrt{6}\right)=\frac{1}{2}[/latex] is equal to [latex]{6}^{\frac{1}{2}}=\sqrt{6}[/latex].
- [latex]{\mathrm{log}}_{3}\left(9\right)=2[/latex] Here, b = 3, y = 2, and x = 9. Therefore, the equation [latex]{\mathrm{log}}_{3}\left(9\right)=2[/latex] is equal to [latex]{3}^{2}=9[/latex].
Try It
Write the following logarithmic equations in exponential form.- [latex]{\mathrm{log}}_{10}\left(1,000,000\right)=6[/latex]
- [latex]{\mathrm{log}}_{5}\left(25\right)=2[/latex]
Answer:
- [latex]{\mathrm{log}}_{10}\left(1,000,000\right)=6[/latex] is equal to [latex]{10}^{6}=1,000,000[/latex]
- [latex]{\mathrm{log}}_{5}\left(25\right)=2[/latex] is equal to [latex]{5}^{2}=25[/latex]
Convert from Exponential to Logarithmic Form
To convert from exponential to logarithmic form, 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: 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]
Answer: 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 equal 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 equal 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 equal to [latex]{\text{log}}_{10}\left(\frac{1}{10,000}\right)=-4[/latex].
Try It
Write the following exponential equations in logarithmic form.- [latex]{3}^{2}=9[/latex]
- [latex]{5}^{3}=125[/latex]
- [latex]{2}^{-1}=\frac{1}{2}[/latex]
Answer:
- [latex]{3}^{2}=9[/latex] is equal to [latex]{\mathrm{log}}_{3}\left(9\right)=2[/latex]
- [latex]{5}^{3}=125[/latex] is equal to [latex]{\mathrm{log}}_{5}\left(125\right)=3[/latex]
- [latex]{2}^{-1}=\frac{1}{2}[/latex] is equal to [latex]{\text{log}}_{2}\left(\frac{1}{2}\right)=-1[/latex]
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- Question ID 29668, 29661. Authored by: McClure,Caren. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.
- College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. Located at: https://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].