Which formula represents the change of base for logarithms logb(a)?

The change of base formula for logarithms allows you to convert a logarithm from one base to another. Specifically, the formula for changing the base of a logarithm from base ( b ) to base ( c ) is expressed as:

[

\log_b(a) = \frac{\log_c(a)}{\log_c(b)}

]

This means that to find the logarithm of ( a ) with respect to ( b ), you can calculate the logarithm of ( a ) with respect to another base ( c ) and divide it by the logarithm of ( b ) with respect to the same base ( c ).

In this case, choosing option B is correct because:

1. It accurately presents the mathematical relationship described by the change of base formula.

2. It illustrates how to convert the logarithmic expression using a different base, which is essential for solving problems that may not readily provide the base required.

The other options do not correctly represent the change of base formula. For example, multiplying logarithmic values or summing logarithmic arguments would not yield the correct relationship between the original base and the new base being used.