Example Simplify following logarithms.

log₄( x³ y⁵ )

Solution

Here the instructions may be a little misleading.  While we say simplify we actually mean to say that we desire to use as many of the logarithm properties as we can.

Note that we can't utilize Property 7 to bring the 3 & the 5 down into the front of the logarithm at this point. To use Property 7 the entire term in the logarithm required to be raised to the power.  In this case the two exponents are just on individual terms in the logarithm and thus Property 7 can't be used here.

However, we do have a product within the logarithm thus we can use Property 5 on this logarithm.

log₄( x³ y⁵ )= log₄  (x ³) + log₄  (y⁵)

Now that we've done it we can utilizes Property 7 on each of these individual logarithms to obtain the final simplified answer.

                       log₄( x³ y⁵ ) = 3 log₄ x + 5 log₄  y