5. \( \log _{7}(3 x+5)=2 \)

To solve the equation \( \log _{7}(3 x+5)=2 \), we start by rewriting it in exponential form. This gives us: \[ 3x + 5 = 7^2 \] Calculating \( 7^2 \) results in: \[ 3x + 5 = 49 \] Next, we isolate \( 3x \) by subtracting 5 from both sides: \[ 3x = 49 - 5 \] \[ 3x = 44 \] Now, divide both sides by 3 to solve for \( x \): \[ x = \frac{44}{3} \] Thus, the solution is: \[ x \approx 14.67 \] To ensure our answer is correct, we can substitute \( x \) back into the original logarithmic equation. After confirming that our value satisfies the equation, we can confidently say \( x = \frac{44}{3} \) is correct!