8b) Solve using logs \( \rightarrow 5(7)^{5 x}=60 \)

To solve the equation \( 5(7)^{5x} = 60 \), first, isolate the exponential term by dividing both sides by 5: \[ (7)^{5x} = \frac{60}{5} = 12 \] Next, apply logarithms to both sides (using natural logarithm or base 10 is fine): \[ \log((7)^{5x}) = \log(12) \] Using the power rule of logarithms, this simplifies to: \[ 5x \log(7) = \log(12) \] Now, solve for \( x \): \[ x = \frac{\log(12)}{5 \log(7)} \] You can plug in your calculator to find the numerical value of \( x \). Happy calculating! --- The use of logarithms dates back to the early 17th century and revolutionized mathematics. They were developed by John Napier for simplifying calculations, especially multiplication and division. Imagine solving complex equations without the convenience of calculators—Napier's invention made life easier for astronomers and navigators back then! If you're interested in diving deeper into logarithms, consider reading "Logarithms: A Brief History" or "The Art of Logarithmic Calculations." These resources can give you a playful insight into how logarithms transformed calculation methods and their application in various fields we encounter today!