# The change of base formula

• May 17th 2009, 06:20 PM
Phire
The change of base formula
I know the definition of the change of base formula, but I really don't know why it's actually called the "change of base" formula. I know that it allows you to solve for an exponent when the original log is not in terms of base 10 or e. For example: log3(5) = log(5)/log(3). Besides solving for the exponent, what exactly did I do here? Where exactly was a base changed? Was this supposed to change the base of the original log? Sorry if this sounds stupid. Here is a definition I read: "A formula that allows you to rewrite a logarithm in terms of logs written with another base." So does this mean that I could change: log4(x) in in terms of base 2? I could write log2^2(x), but I don't think that would help much? If I used the change of base formula, it would be: log2(x)/log2(4), which just creates a fraction.
• May 17th 2009, 07:41 PM
Shyam
Quote:

Originally Posted by Phire
I know the definition of the change of base formula, but I really don't know why it's actually called the "change of base" formula. I know that it allows you to solve for an exponent when the original log is not in terms of base 10 or e. For example: log3(5) = log(5)/log(3). Besides solving for the exponent, what exactly did I do here? Where exactly was a base changed? Was this supposed to change the base of the original log? Sorry if this sounds stupid. Here is a definition I read: "A formula that allows you to rewrite a logarithm in terms of logs written with another base." So does this mean that I could change: log4(x) in in terms of base 2? I could write log2^2(x), but I don't think that would help much? If I used the change of base formula, it would be: log2(x)/log2(4), which just creates a fraction.

$\log_4 x = \frac{\log_2 x}{\log_2 4}$

$= \frac{\log_2 x}{\log_2 (2^2)}$

$= \frac{\log_2 x}{2\log_2 2}$

$= \frac{1}{2} \log_2 x$

$= \log_2 x^{\frac{1}{2}}$

$\log_4 x = \log_2 \sqrt x$
• May 18th 2009, 09:13 AM
stapel
Quote:

Originally Posted by Phire
So does this mean that I could change: log4(x) in in terms of base 2? ...If I used the change of base formula, it would be: log2(x)/log2(4), which just creates a fraction.

Yes, but that fraction contains logs having a base of 2, where you'd started with one log having a base of 4. The base has been changed. (Wink)
• May 18th 2009, 09:28 AM
e^(i*pi)
Quote:

Originally Posted by Phire
If I used the change of base formula, it would be: log2(x)/log2(4), which just creates a fraction.

Some calculators only have base 10 (log) and base e (ln) buttons so the change of base formula can also let your calculator solve it