18. \( 2 \cdot 3^{x-4}-1=15 \)

To solve the equation \( 2 \cdot 3^{x-4} - 1 = 15 \), start by isolating the exponential term. Add 1 to both sides: \[ 2 \cdot 3^{x-4} = 16 \] Now, divide both sides by 2: \[ 3^{x-4} = 8 \] Next, express \( 8 \) as a power of \( 3 \). Since \( 8 = 2^3 \), we can express \( 3^{x-4} \) in logarithmic terms. Take the logarithm of both sides: \[ x - 4 = \log_3(8) \] Now, add \( 4 \) to both sides for the final result: \[ x = \log_3(8) + 4 \] Since \( \log_3(8) \) doesn't simplify nicely, you might want to calculate its approximate value using change of base formula: \[ x \approx \frac{\log(8)}{\log(3)} + 4 \] Which gives you a numerical approximation for the value of \( x \). Calculating the approximate value yields: \[ x \approx 4 + 1.8928 \] \[ x \approx 5.8928 \] You can round it to \( x \approx 5.89 \). Happy solving!