Which identity is true for logarithms with bases x and y where x,y > 0 and not equal to 1?

The identity that is true for logarithms with bases ( x ) and ( y ) where ( x, y > 0 ) and not equal to 1 is that the logarithm of a product can be expressed as the sum of the logarithms of the individual factors.

This property is stated mathematically as ( \log_b(xy) = \log_b(x) + \log_b(y) ). This identity holds because logarithms can be thought of as the inverse operations of exponentiation, and when multiplying the arguments inside a logarithm, it is equivalent to adding their respective exponents when converted to the base ( b ).

The function of logarithms allows for translation of multiplication in the input (the arguments being evaluated) to addition in the output (the value returned by the logarithm). This is fundamental in simplifying expressions and solving equations involving logarithms.

Other identities presented, such as those involving division or other operations, do not uphold the same property where multiplication in arguments corresponds directly to addition in their logarithmic forms. Therefore, acknowledging this crucial aspect of logarithmic functions allows a better understanding and application of their properties.