Solve for \( m \) \( 4 \times 6^{m}=6 \times 4^{3} \)

To solve for \( m \) in the equation \( 4 \times 6^{m} = 6 \times 4^{3} \), we'll start by simplifying the right side. First, calculate \( 4^{3} \): \[ 4^{3} = 64 \] Now substitute this back into the equation: \[ 4 \times 6^{m} = 6 \times 64 \] Next, calculate \( 6 \times 64 \): \[ 6 \times 64 = 384 \] Now our equation looks like: \[ 4 \times 6^{m} = 384 \] Now, divide both sides by 4: \[ 6^{m} = \frac{384}{4} = 96 \] Next, we can express \( 96 \) as a power of \( 6 \) or rewrite it. Our goal is to use logarithms to solve for \( m \). We know that: \[ m = \log_{6}(96) \] Using the change of base formula, this can be rewritten (using base 10 or natural logarithm): \[ m = \frac{\log(96)}{\log(6)} \] We can approximate these values using a calculator: \[ \log(96) \approx 1.9823 \quad \text{and} \quad \log(6) \approx 0.7781 \] Thus, we approximate: \[ m \approx \frac{1.9823}{0.7781} \approx 2.545 \] So the solution for \( m \) is approximately: \[ m \approx 2.55 \]