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Course: Algebra 2 > Unit 8
Lesson 3: The change of base formula for logarithms- Evaluating logarithms: change of base rule
- Logarithm change of base rule intro
- Evaluate logarithms: change of base rule
- Using the logarithm change of base rule
- Use the logarithm change of base rule
- Proof of the logarithm change of base rule
- Logarithm properties review
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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 . Since is not a rational power of , it is difficult to evaluate this without a calculator.
However, most calculators only directly calculate logarithms in base- and base- . So in order to find the value of , 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
. - As always, the arguments of the logarithms must be positive and the bases of the logarithms must be positive and not equal to
in order for this property to hold!
Example: Evaluating
If your goal is to find the value of a logarithm, change the base to or since these logarithms can be calculated on most calculators.
So let's change the base of to .
To do this, we apply the change of base rule with , , and .
We can now find the value using the calculator.
Check your understanding
Justifying the change of base rule
At this point, you might be thinking, "Great, but why does this rule work?"
Let's start with a concrete example. Using the above example, we want to show that .
Let's use as a placeholder for . In other words, we have . From the definition of logarithms it follows that . Now we can perform a sequence of operations on both sides of this equality so the equality is maintained:
Since was defined to be , we have that as desired!
By the same logic, we can prove the change of base rule. Just change to , to and pick any base as the new base, and you have your proof!
Challenge problems
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