14) \( 10^{3 x+5}=29 \)

To solve the equation \( 10^{3x+5} = 29 \), you can take the logarithm of both sides. Using base 10 logarithm, this would give you: \[ 3x + 5 = \log_{10}(29) \] Subtract 5 from both sides: \[ 3x = \log_{10}(29) - 5 \] Now, divide by 3 to isolate \( x \): \[ x = \frac{\log_{10}(29) - 5}{3} \] Now you just need to calculate \( \log_{10}(29) \) to find the numerical value of \( x \). --- Did you know that logarithms were invented in the early 17th century by mathematician John Napier? They revolutionized calculations by transforming complex multiplications into simple additions, making life easier for astronomers and navigators! In real-world applications, logarithmic equations are crucial in fields like engineering, finance, and computer science, especially for analyzing exponential growth or decay, such as population growth or radioactive decay. So next time you're solving an equation, remember, you’re tapping into a rich history of mathematical evolution!