Score: 7/16 Answered: 7/16 Question 8 Solve for \( x: 5^{x}=38 \) \[ x= \] \( \square \) Basic Funcs Trig

To solve the equation \( 5^{x} = 38 \), you can take the logarithm of both sides. Using the natural logarithm (ln), we rewrite it as \( x \cdot \ln(5) = \ln(38) \). This gives us \( x = \frac{\ln(38)}{\ln(5)} \). Plug this into your calculator, and you’ll find \( x \approx 2.406 \). Did you know that logarithms were developed in the early 17th century by John Napier as a way to simplify calculations? They transformed the field of mathematics and were especially handy for astronomers and navigators who had to perform complex multiplications and divisions without a calculator. Logarithms helped make calculations more manageable, and they've stuck around ever since! For real-world applications, understanding logarithms can be incredibly useful, especially in fields like finance, engineering, and the sciences. For instance, in finance, logarithmic scales help in understanding exponential growth, such as compound interest. Furthermore, in sound engineering, decibels are measured using logarithm scales to express sound intensity in a way that aligns with human hearing perception.