Question 379858
{{{log(2, (log(b, (sqrt(b)))))}}}
You are correct. start by rewriting {{{sqrt(b)}}} as {{{b^(1/2)}}}:
{{{log(2, (log(b, (b^(1/2)))))}}}
Next, on the inner logarithm, we can use a property of logarithms, {{{log(a, (p^q)) = q*log(a, (p))}}}, to "move" the exponent of the argument out in front of the logarithm:
{{{log(2, ((1/2)*log(b, (b))))}}}
By definition, {{{log(b, (b)) = 1}}} so this becomes:
{{{log(2, (1/2))}}}
The quick way to finish requires that we understand what this expression represents. It represents the exponent for 2 that results in 1/2. If you also understand exponents, you will know that the answer is: -1!<br>
The long way is to use another property of logarithms, {{{log(a, (p/q)) = log(a, (p)) - log(a, (q))}}}, to split the numerator and denominator into separate logarithms:
{{{log(2, (1)) - log(2, (2))}}}
The logarithm of 1, regardless of the base, is zero (because any number to the zero power is 1). And by definition {{{log(2, (2)) = 1}}}. So now we have:
0 - 1
which equals
-1