Which logarithmic identity represents the change of base formula?

The change of base formula is a key identity in logarithms that allows one to convert logarithms from one base to another. This is particularly useful when a calculator or computational tool is only able to compute logarithms in a specific base, typically base 10 or base e (natural logarithm).

The change of base formula states that the logarithm of a number ( a ) in base ( b ) can be expressed using a different base ( k ) as follows: ( \log_b(a) = \frac{\log_k(a)}{\log_k(b)} ). This identity helps students understand how logarithms in different bases relate to one another, making it easier to calculate or manipulate logarithmic expressions.

When you apply this formula, you take the logarithm of ( a ) in base ( k ) and divide it by the logarithm of ( b ) in the same base ( k ). This division effectively allows you to express ( \log_b(a) ) without needing to work directly in base ( b ).

This formula is central to many applications of logarithms, including solving equations and simplifying logarithmic expressions in various contexts.