Question 245805
To solve for variables in the argument of a logarithm we generally rewrite the logarithmic equation in exponential form. Since your equation has a logarithm of a logarithm of a logarithm, we will have to do this three times.<br>
Rewriting logarithmic equations in exponential form requires knowing that
{{{log(a, (p)) = q}}} is equivalent to {{{p = a^q}}}
The "a", "p" and "q" can be any expression. We will use this to "peel away" the logarithms, one at a time, like peeling the layers of an onion:
{{{ ln(log(2, (ln(x)))) = 0}}}
In exponential form:
{{{ log(2, (ln(x))) = e^0}}} (Since the base of ln is e.)
Since any non-zero number, including e, to the zero power is 1 this simplifies to:
{{{ log(2, (ln(x))) = 1}}}
This equation in exponential form:
{{{ln(x) = 2^1}}}
Since 2^1 = 2 this simplifies to:
{{{ln(x) = 2}}}
This equation in exponential form:
{{{x = e^2}}}