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Guias de estudo > Intermediate Algebra

Read: Change of Base

Learning Objectives

  • Use properties of logarithms to define the change of base formula
  • Change the base of logarithmic expressions into base 10, or base e

Most calculators can evaluate only common and natural logs. In order to evaluate logarithms with a base other than 1010 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 n≠1n\ne 1 and b≠1b\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. By taking the log base nn of both sides of the equation, we arrive at an exponential form, namely 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}{c}{\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}

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}{c}{\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}
Let's practice changing the base of a logarithmic expression from 55 to base e.

Example

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, n =en = 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 33. The denominator of the quotient will be the natural log with argument 55.

logbM=lnMlnblog53=ln3ln5\begin{array}{c}{\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}

We can generalize the change of base formula in the following way:

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 n≠1n\ne 1 and b≠1b\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}
As we stated earlier, the main reason for changing the base of a logarithm is to be able to evaluate it with a calculator. In the following example we will use the change of base formula on a logarithmic expression, then evaluate the result with a calculator.

Example

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 22 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}{c}{\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}

 

Think About It

Can we change common logarithms to natural logarithms? Write your ideas in the textbox below before looking at the solution. [practice-area rows="1"][/practice-area]

Answer: 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}.

 Summary

For practical purposes found in many different sciences or finance applications, you may want to evaluate a logarithm with a calculator. The change of base formula will allow you to change the base of any logarithm to either 1010 or e  so you can evaluate it with a calculator. Here we have summarized the steps for using the change of base formula.

Given a logarithm with the form logbM{\mathrm{log}}_{b}M

  1. Determine the new base n, remembering that the common log, log(x)\mathrm{log}\left(x\right), has base 1010, 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.

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