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Study Guides > College Algebra CoRequisite Course

Change of Base Formula

Learning Outcomes

  • Rewrite logarithms with a different base using the change of base formula.

Using the Change-of-Base Formula for Logarithms

Most calculators can only evaluate common and natural logs. In order to evaluate logarithms with a base other than 10 or ee, we use the change-of-base formula to rewrite the logarithm as the quotient of logarithms of any other base; when using a calculator, we would change them to common or natural logs. To derive the change-of-base formula, we use the one-to-one property and power rule for logarithms. Given any positive real numbers M, b, and n, where n1n\ne 1 and b1b\ne 1, we show

logbM=lognMlognb{\mathrm{log}}_{b}M\text{=}\frac{{\mathrm{log}}_{n}M}{{\mathrm{log}}_{n}b}

Let y=logbMy={\mathrm{log}}_{b}M. Converting to exponential form, we obtain by=M{b}^{y}=M. It follows that:

logn(by)=lognMApply the one-to-one property.ylognb=lognMApply the power rule for logarithms.y=lognMlognbIsolate y.logbM=lognMlognbSubstitute for y.\begin{array}{l}{\mathrm{log}}_{n}\left({b}^{y}\right)\hfill & ={\mathrm{log}}_{n}M\hfill & \text{Apply the one-to-one property}.\hfill \\ y{\mathrm{log}}_{n}b\hfill & ={\mathrm{log}}_{n}M \hfill & \text{Apply the power rule for logarithms}.\hfill \\ y\hfill & =\frac{{\mathrm{log}}_{n}M}{{\mathrm{log}}_{n}b}\hfill & \text{Isolate }y.\hfill \\ {\mathrm{log}}_{b}M\hfill & =\frac{{\mathrm{log}}_{n}M}{{\mathrm{log}}_{n}b}\hfill & \text{Substitute for }y.\hfill \end{array}

applying the one-to-one property of logarithms and exponents

In the demonstration above, deriving the change-of-base formula from the definition of the logarithm, we applied the one-to-one property of logarithms to

by=Mb^y=M

to obtain

lognby=lognM\log_nb^y=\log_nM.

The application of the property is sometimes referred to as a property of equality with regard to taking the log base nn on both sides, where nn is any real number.  Recall that the one-to-one property states that logbM=logbNM=N\log_bM=\log_bN \Leftrightarrow M=N. We take the double-headed arrow to mean if and only if and use it when the equality can be implied in either direction. Therefore, it is just as appropriate to state that M=NlogbM=logbNM=N \Leftrightarrow \log_bM=\log_bN, which is what we did in the derivation above. That is, by=Mlognby=lognMb^y=M \Leftrightarrow \log_nb^y=\log_nM. The same idea applies to the one-to-one property of exponents. Since am=anm=na^m=a^n \Leftrightarrow m=n, it is also true that given m=nm=n, we can write qm=qnq^m=q^n for qq, any real number. This idea leads to important techniques for solving logarithmic and exponential equations. Keep it in mind as you work through the rest of the module.
For example, to evaluate log536{\mathrm{log}}_{5}36 using a calculator, we must first rewrite the expression as a quotient of common or natural logs. We will use the common log.

log536=log(36)log(5)Apply the change of base formula using base 10.2.2266 Use a calculator to evaluate to 4 decimal places.\begin{array}{l}{\mathrm{log}}_{5}36\hfill & =\frac{\mathrm{log}\left(36\right)}{\mathrm{log}\left(5\right)}\hfill & \text{Apply the change of base formula using base 10}\text{.}\hfill \\ \hfill & \approx 2.2266\text{ }\hfill & \text{Use a calculator to evaluate to 4 decimal places}\text{.}\hfill \end{array}

A General Note: The Change-of-Base Formula

The change-of-base formula can be used to evaluate a logarithm with any base. For any positive real numbers M, b, and n, where n1n\ne 1 and b1b\ne 1,

logbM=lognMlognb{\mathrm{log}}_{b}M\text{=}\frac{{\mathrm{log}}_{n}M}{{\mathrm{log}}_{n}b}.

It follows that the change-of-base formula can be used to rewrite a logarithm with any base as the quotient of common or natural logs.

logbM=lnMlnb{\mathrm{log}}_{b}M=\frac{\mathrm{ln}M}{\mathrm{ln}b}

and

logbM=logMlogb{\mathrm{log}}_{b}M=\frac{\mathrm{log}M}{\mathrm{log}b}

How To: Given a logarithm Of the form logbM{\mathrm{log}}_{b}M, use the change-of-base formula to rewrite it as a quotient of logs with any positive base nn, where n1n\ne 1

  1. Determine the new base n, remembering that the common log, log(x)\mathrm{log}\left(x\right), has base 10 and the natural log, ln(x)\mathrm{ln}\left(x\right), has base e.
  2. Rewrite the log as a quotient using the change-of-base formula:
    • The numerator of the quotient will be a logarithm with base n and argument M.
    • The denominator of the quotient will be a logarithm with base n and argument b.

Example: Changing Logarithmic Expressions to Expressions Involving Only Natural Logs

Change log53{\mathrm{log}}_{5}3 to a quotient of natural logarithms.

Answer: Because we will be expressing log53{\mathrm{log}}_{5}3 as a quotient of natural logarithms, the new base = e. We rewrite the log as a quotient using the change-of-base formula. The numerator of the quotient will be the natural log with argument 3. The denominator of the quotient will be the natural log with argument 5.

logbM=lnMlnblog53=ln3ln5\begin{array}{l}{\mathrm{log}}_{b}M\hfill & =\frac{\mathrm{ln}M}{\mathrm{ln}b}\hfill \\ {\mathrm{log}}_{5}3\hfill & =\frac{\mathrm{ln}3}{\mathrm{ln}5}\hfill \end{array}

tip for success

Even if your calculator has a logarithm function for bases other than 1010 or ee, you should become familiar with the change-of-base formula. Being able to manipulate formulas by hand is a useful skill in any quantitative or STEM-related field.

Try It

Change log0.58{\mathrm{log}}_{0.5}8 to a quotient of natural logarithms.

Answer: ln8ln0.5\frac{\mathrm{ln}8}{\mathrm{ln}0.5}

[ohm_question]86013[/ohm_question]

Q & A

Can we change common logarithms to natural logarithms? Yes. Remember that log9\mathrm{log}9 means log109{\text{log}}_{\text{10}}\text{9}. So, log9=ln9ln10\mathrm{log}9=\frac{\mathrm{ln}9}{\mathrm{ln}10}.

Example: Using the Change-of-Base Formula with a Calculator

Evaluate log2(10){\mathrm{log}}_{2}\left(10\right) using the change-of-base formula with a calculator.

Answer: According to the change-of-base formula, we can rewrite the log base 2 as a logarithm of any other base. Since our calculators can evaluate the natural log, we might choose to use the natural logarithm which is the log base e. log210=ln10ln2Apply the change of base formula using base e.3.3219Use a calculator to evaluate to 4 decimal places.\begin{array}{l}{\mathrm{log}}_{2}10=\frac{\mathrm{ln}10}{\mathrm{ln}2}\hfill & \text{Apply the change of base formula using base }e.\hfill \\ \approx 3.3219\hfill & \text{Use a calculator to evaluate to 4 decimal places}.\hfill \end{array}

Try It

Evaluate log5(100){\mathrm{log}}_{5}\left(100\right) using the change-of-base formula.

Answer: ln100ln54.60511.6094=2.861\frac{\mathrm{ln}100}{\mathrm{ln}5}\approx \frac{4.6051}{1.6094}=2.861

[ohm_question]35015[/ohm_question]

Try it

The first graphing calculators were programmed to only handle logarithms with base 10. One clever way to create the graph of a logarithm with a different base was to change the base of the logarithm using the principles from this section. Use an online graphing tool to plot f(x)=log10xlog102f(x)=\frac{\log_{10}{x}}{\log_{10}{2}}. Follow these steps to see a clever way to graph a logarithmic function with base other than 10 on a graphing tool that only knows base 10.
  • Enter the function g(x)=log2xg(x) = \log_{2}{x}
  • Can you tell the difference between the graph of this function and the graph of f(x)f(x)? Explain what you think is happening.
  • Your challenge is to write two new functions h(x), and k(x)h(x),\text{ and }k(x) that include a slider so you can change the base of the functions. Remember that there are restrictions on what values the base of a logarithm can take. You can click on the endpoints of the slider to change the input values.

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