Question 155459
First, you recall that the definition of the logarithm of a numer is:
"The logarithm of a number is the power to which the base of the logarithm must be raised to equal the number"
a) {{{Log[16](512) = x}}} so, by the above definition, x is the power to which 16 (the base) must be raised to equal the number (512). So you can write that out as described:
{{{16^x = 512}}} The next step is to exprees the the numbers on both sides as powers of the same base.
{{{16 = 2^4}}} and {{{512 = 2^9}}} so make the appropriate substitutions to get:
{{{2^(4x) = 2^9}}} now since the bases (2) are equal, the exponents are equal, so...
{{{4x = 9}}} and
{{{x = 9/4}}} so
{{{Log[16](512) = 9/4}}} or
{{{Log[16](512) = 2.25}}}
Similarly:
b) {{{Log[32](64) = x}}}
{{{32^x = 64}}} but{{{32 = 2^5}}} and {{{64 = 2^6}}} so...
{{{2^5x = 2^6}}} so...
{{{5x = 6}}} and..
{{{x = 6/5}}} then...
{{{Log[16](64) = 6/5}}} or 
{{{Log[16](64) = 1.2}}}