Question: Solve the following logarithmic equation: \[ \log_4(4)^{\log_2\left(\frac{1}{4}\right)} = x…

Solve the following logarithmic equation:

\[ \log_4(4)^{\log_2\left(\frac{1}{4}\right)} = x \]

Your answer (opens in new window)

\[ x = \boxed{\phantom{0}} \]

Solution

To solve the given equation: \[ \log_4(4)^{\log_2\left(\frac{1}{4}\right)} = x \] First, simplify \(\log_4(4)\). \(\log_4(4) = 1\) because \(4^1 = 4\). Substitute back into the equation: \[ 1^{\log_2\left(\frac{1}{4}\right)} = x \] Since any number raised to any power is still that number if it’s 1: \[ 1 = x \] Thus, the solution is: \[ x = 1 \]