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## Algebra 2

### Course: Algebra 2>Unit 8

Lesson 4: The change of base formula for logarithms

# Logarithm change of base rule intro

Learn how to rewrite any logarithm using logarithms with a different base. This is very useful for finding logarithms in the calculator!
Suppose we wanted to find the value of the expression log, start base, 2, end base, left parenthesis, 50, right parenthesis. Since 50 is not a rational power of 2, it is difficult to evaluate this without a calculator.
However, most calculators only directly calculate logarithms in base-10 and base-e. So in order to find the value of log, start base, 2, end base, left parenthesis, 50, right parenthesis, we must change the base of the logarithm first.

## The change of base rule

We can change the base of any logarithm by using the following rule:
Notes:
• When using this property, you can choose to change the logarithm to any base start color #0d923f, x, end color #0d923f.
• As always, the arguments of the logarithms must be positive and the bases of the logarithms must be positive and not equal to 1 in order for this property to hold!

## Example: Evaluating $\log_2(50)$log, start base, 2, end base, left parenthesis, 50, right parenthesis

If your goal is to find the value of a logarithm, change the base to 10 or e since these logarithms can be calculated on most calculators.
So let's change the base of log, start base, 2, end base, left parenthesis, 50, right parenthesis to start color #1fab54, 10, end color #1fab54.
To do this, we apply the change of base rule with b, equals, 2, a, equals, 50, and x, equals, 10.
\begin{aligned}\log_\blueD{2}(\purpleC{50})&=\dfrac{\log_{\greenD{10}}(\purpleC{50})}{\log_{\greenD{10}}(\blueD2)} &&{\gray{\text{Change of base rule}}} \\\\ &=\dfrac{\log(50)}{\log(2)} &&{\gray{\text{Since} \log_{10}(x)=\log(x)}} \end{aligned}
We can now find the value using the calculator.
start fraction, log, left parenthesis, 50, right parenthesis, divided by, log, left parenthesis, 2, right parenthesis, end fraction, approximately equals, 5, point, 644

Problem 1
Evaluate log, start base, 3, end base, left parenthesis, 20, right parenthesis.

Problem 2
Evaluate log, start base, 7, end base, left parenthesis, 400, right parenthesis.

Problem 3
Evaluate log, start base, 4, end base, left parenthesis, 0, point, 3, right parenthesis.

## Justifying the change of base rule

At this point, you might be thinking, "Great, but why does this rule work?"
log, start base, b, end base, left parenthesis, a, right parenthesis, equals, start fraction, log, start base, x, end base, left parenthesis, a, right parenthesis, divided by, log, start base, x, end base, left parenthesis, b, right parenthesis, end fraction
Let's start with a concrete example. Using the above example, we want to show that log, start base, 2, end base, left parenthesis, 50, right parenthesis, equals, start fraction, log, left parenthesis, 50, right parenthesis, divided by, log, left parenthesis, 2, right parenthesis, end fraction.
Let's use n as a placeholder for log, start base, 2, end base, left parenthesis, 50, right parenthesis. In other words, we have log, start base, 2, end base, left parenthesis, 50, right parenthesis, equals, n. From the definition of logarithms it follows that 2, start superscript, n, end superscript, equals, 50. Now we can perform a sequence of operations on both sides of this equality so the equality is maintained:
\begin{aligned} 2^n &= 50 \\\\ \log(2^n) &= \log(50)&&{\gray{\text{If }A=B\text{, then }\log(A)=\log(B)}} \\\\ n\log(2)&=\log(50)&&{\gray{\text{Power Rule}}} \\\\ n &= \dfrac{\log(50)}{\log(2)} &&{\gray{\text{Divide both sides by} \log(2)}} \end{aligned}
Since n was defined to be log, start base, 2, end base, left parenthesis, 50, right parenthesis, we have that log, start base, 2, end base, left parenthesis, 50, right parenthesis, equals, start fraction, log, start base, x, end base, left parenthesis, 50, right parenthesis, divided by, log, start base, x, end base, left parenthesis, 2, right parenthesis, end fraction as desired!
By the same logic, we can prove the change of base rule. Just change 2 to b, 50 to a and pick any base x as the new base, and you have your proof!

## Challenge problems

Challenge problem 1
Evaluate start fraction, log, left parenthesis, 81, right parenthesis, divided by, log, left parenthesis, 3, right parenthesis, end fraction without using a calculator.

Challenge problem 2
Which expression is equivalent to log, left parenthesis, 6, right parenthesis, dot, log, start base, 6, end base, left parenthesis, a, right parenthesis?