Question: Solve the following logarithmic equation: \[ \log_8(4) = x \] Enter your answer (opens in new…

Solve the following logarithmic equation:

\[ \log_8(4) = x \]

Enter your answer (opens in new window)

\[ x = \boxed{\quad} \]

Solution

To solve the logarithmic equation \( \log_8(4) = x \), we can convert it from logarithmic form to exponential form. The equation \( \log_8(4) = x \) means that: \[ 8^x = 4 \] Next, express both numbers as powers of the same base. We know that: \[ 8 = 2^3 \] \[ 4 = 2^2 \] Substitute these into the equation: \[ (2^3)^x = 2^2 \] Simplify the left side of the equation: \[ 2^{3x} = 2^2 \] Since the bases are the same, you can set the exponents equal to each other: \[ 3x = 2 \] Solve for \( x \): \[ x = \frac{2}{3} \] The value of \( x \) is \( \frac{2}{3} \).