Question 238662
{{{log(4, (log(4, (x)))) = - 4}}}
Solving equations of the form
log(some-expression-with-a-variable) = another-expression
is usually done by rewriting the equaiton in exponential form. To rewrite logarithmic equations in logarithmic form we need to remember that {{{log(a, (p)) = q}}} is equivalent to {{{p = a^q}}}<br>
Since your equation has a logarithm within a logarithm, we will have to do this twice. Rewriting the outer logarithm in exponential form we get:
{{{log(4, (x)) = 4^(-4)}}}
Since {{{4^(-4) = 1/(4^4)}}} and {{{4^4 = 4*4*4*4 = 256}}}, the right side simplifies to:
{{{log(4, (x)) = 1/256}}}
Now we will rewrite the remaining logarithm in exponential form:
{{{x = 4^(1/256)}}}
This may be an acceptable form for the answer. But it can be simplified a little. First we can "reduce" the exponent using a bit of cleverness and a good understanding of fractional exponents:
Factor the exponent:
{{{x = 4^((1/2)(1/128))}}}
Use the property {{{a^(p*q) = (a^p)^q}}} to rewrite the abvoe as a power of a power:
{{{x = (4^(1/2))^(1/128)}}}
Since {{{4^(1/2) = sqrt(4)}}} and {{{sqrt(4) = 2}}}, we can replace {{{4^(1/2)}}} with 2. (This is why we factored 1/2 out of the exponent.)
{{{x = 2^(1/128)}}}
This is a simplified form of the answer. With a fractional exponent, we could write this in radical form:
{{{x = root(128, 2)}}}