What is the change of base formula for logarithms?

The change of base formula for logarithms is an essential tool that allows you to compute logarithms in a different base. This formula states that the logarithm of a number ( M ) in base ( b ) can be expressed in terms of logarithms in another base ( c ). Specifically, the formula is:

[

\log_b M = \frac{\log_c M}{\log_c b}

]

This means that to find the logarithm of ( M ) in base ( b ), you take the logarithm of ( M ) in base ( c ) and divide it by the logarithm of ( b ) in base ( c ). This is particularly useful when your calculator only has a logarithm function for a specific base (commonly base 10 or base ( e )), allowing you to compute logarithms of any base with accuracy.

For instance, if you want to find ( \log_2 8 ), you can use the change of base formula by choosing base 10 (or any base that you prefer) to perform the calculation as follows:

[

\log_2 8 = \frac{\log_{10} 8}{\log_{10}