Question 240807
{{{log(8, (sqrt(8^9))) = n}}}
As usual, there are several ways to solve this. Maybe the quickest way is based on<ul><li>knowing our exponents (including fractional exponents) well</li><li>seeing that the argument of the base 8 logarithm is a power of 8</li><li>knowing that you do not need a calculator to find the base 8 logarithm of a power of 8</li></ul>
Since square roots are the same as raising to the 1/2 power we can rewrite the argument:
{{{log(8, ((8^9)^(1/2))) = n}}}
Using the property of exponents, {{{(a^p)^q = a^(p*q)}}}, we get:
{{{log(8, (8^(9/2))) = n}}}
Using the property of logarithms, {{{log(a, (p^q)) = q*log(a, (p))}}}, we can move the exponent out in front:
{{{(9/2)log(8, (8)) = n}}}
Since by definition {{{log(a, (a)) = 1}}}:
{{{9/2 = n}}}