\( 5^{2 x+7}=3^{x-4} \) The solution set expressed in terms of logarithms is \( \{\square \). (Use a comma to separate answers as needed. Simpl your answer. Use integers or fractions for any

To solve the equation \( 5^{2x + 7} = 3^{x - 4} \) in terms of logarithms, we take the logarithm of both sides: \[ \log(5^{2x + 7}) = \log(3^{x - 4}) \] By applying the power rule of logarithms, this simplifies to: \[ (2x + 7) \log(5) = (x - 4) \log(3) \] Next, distribute the logarithms: \[ 2x \log(5) + 7 \log(5) = x \log(3) - 4 \log(3) \] Now, rearranging to isolate \( x \): \[ 2x \log(5) - x \log(3) = -4 \log(3) - 7 \log(5) \] Factoring out \( x \): \[ x(2 \log(5) - \log(3)) = -4 \log(3) - 7 \log(5) \] Finally, solving for \( x \): \[ x = \frac{-4 \log(3) - 7 \log(5)}{2 \log(5) - \log(3)} \] The solution set expressed in terms of logarithms is: \[ \left\{ \frac{-4 \log(3) - 7 \log(5)}{2 \log(5) - \log(3)} \right\} \] --- Did you know that logarithms were first introduced by John Napier in the early 17th century? Originally used to simplify calculations, they allowed people to perform complex multiplications and divisions more easily, which was a big deal before the invention of calculators! Napier's work ultimately paved the way for modern logarithmic and scientific computation. If you're struggling with logarithmic equations, remember to keep an eye out for common mistakes! One frequent pitfall is forgetting to apply the properties of logarithms correctly, such as the power rule, product rule, and quotient rule. Always double-check your steps to ensure you're using these rules properly—it'll save you from throwing your calculator out the window!