Algebra
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Boundless Algebra
Exponents, Logarithms, and Inverse Functions
Working With Logarithms
Algebra Textbooks Boundless Algebra Exponents, Logarithms, and Inverse Functions Working With Logarithms
Algebra Textbooks Boundless Algebra Exponents, Logarithms, and Inverse Functions
Algebra Textbooks Boundless Algebra
Algebra Textbooks
Algebra
Concept Version 9
Created by Boundless

Changing Logarithmic Bases

A logarithm written in one base can be converted to an equal quantity written in a different base. 

Learning Objective

  • Use the change of base formula to convert logarithms to different bases


Key Points

    • The base of a logarithm can be changed by expressing it as the quotient of two logarithms with a common base.
    • Changing a logarithm's base to $10$ makes it much simpler to evaluate; it can be done on a calculator.

Terms

  • base

    A number raised to the power of an exponent.

  • logarithm

    The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number.


Full Text

Most common scientific calculators have a key for computing logarithms with base $10$, but do not have keys for other bases. So, if you needed to get an approximation to a number like $\log_4(9)$ it can be difficult to do so. One could easily guess that it is between $1$ and $2$ since $9$ is between $4^1$ and $4^2$, but it is difficult to get an accurate approximation. Fortunately, there is a change of base formula that can help.

Change of Base Formula

The change of base formula for logarithms is:

$\displaystyle \log_a(x)=\frac{\log_b(x)}{\log_b(a)}$

Thus, for example, we could calculate that $\log_4(9)=\frac{\log_{10}(9)}{\log_{10}(4)}$ which could be computed on almost any handheld calculator. 

Deriving the Formula

To see why the formula is true, give $\log_a(x)$ a name like $z$:

$\displaystyle z=\log_a(x)$ 

Write this as $a^z=x$

Now take the logarithm with base $b$ of both sides, yielding:

$\displaystyle \log_b a^z = \log_bx$ 

Using the power rule gives:

 $\displaystyle z \cdot \log_ba = \log_b x$

Dividing both sides by $\log_ba$ gives:

 $\displaystyle z={\log_b x \over \log_ba}.$

Thus we have $\log_a x ={\log_b x \over \log_b a}. $

Example

An expression of the form $\log_5(10^{x^2+1})$ might be easier to graph on a graphing calculator or other device if it were written in base $10$ instead of base 5. The change-of-base formula can be applied to it:

$\displaystyle \log_5(10^{x^2+1}) = {\log_{10}(10^{x^2+1}) \over \log_{10}5}$ 

Which can be written as ${x^2+1 \over \log_{10} 5}. $

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