What is the method of multiplying logarithms with different bases?

In order to multiply logarithms with different bases, the change of base formula must be used. The change of base formula is log_b_a = (log_c_a) / (log_c_b). The easiest way to use the change of base formula is to change the bases to base 10.

1. Use the change of base formula

   With the change of base formula, the logarithms are converted from their present form to a form where they have similar bases so that mathematical operations can be performed on them. If the problem given is log_2_9 * log_3_8, the expression becomes ((log_9) / (log_2)) * ((log_8) / (log_3)). With base 10, the 10 is implicit and doesn't need to be shown.

2. Reduce any argument to an exponent if possible

   The goal is to cancel out any like terms. In some cases, it is necessary to convert larger numbers to exponent form in order to cancel as many terms as possible. (log_9) / (log_2) becomes (log_3^2) / (log_2) and (log_8) / (log_3) becomes (log_2^3) / (log_3).

3. Bring any exponents to the front of each expression

   Any exponents that appear due to conversion will be brought to the front of each expression. The equation becomes (2 * (log_3) / (log_2)) * (3 * (log_2) / (log_3)).

4. Cancel any like terms and simplify

   Any like terms cancel out and whatever is left is simplified. Log_3 cancels out with log_3 and log_2 cancels with log_2. The end equation is 2 * 3, and the answer is 6.