17Calculus Precalculus - Change of Base For Logarithms

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Logarithms Change of Base

As a precalculus/calculus instructor, I am not a fan of this formula. In fact, when I teach, I tell my students that they may not use this formula as part of their work for credit. However, perhaps some of you are allowed to use it or you want another way to check your work. If you are learning it for those reasons, here it is. As usual, always check with your instructor to see what they require.

It is always good to know where your equations and formulas come from. Here is a video proving the change of base formula. It is really easy to follow and understand.

Eddie Woo - Proving "Change of Base" Rule [3min-17secs]

video by Eddie Woo

Change of Base Formula (General)

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

So why do we need the change of base formula and when do we use it? There are two main times it may be helpful. First, not all logarithms can be entered into a calculator. Most calculators can calculate only base 10 and the natural logarithm. So in order to get a value with another base, some people use this formula.
The other reason is that, when you get calculus, you will need to have your expressions using the natural log in order to apply a technique. So you would need to convert a logarithm with another base to the natural log. Here is the change of base formula with the natural log instead of log base b.

Change of Base Formula (Natural Log)

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

Notice that the term in the denominator is just a number, assuming that $$a$$ is a number.

So that's it. There is nothing fancy going on. It's just a basic formula.

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

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