(d) \( 4^{x}=5 \) \( x=\square \)

To solve the equation \( 4^{x}=5 \), you can take the logarithm of both sides. Let's use natural logarithms for this example. The equation becomes \( \ln(4^x) = \ln(5) \). Using the power rule of logarithms, this simplifies to \( x \cdot \ln(4) = \ln(5) \). Therefore, you can isolate \( x \) by dividing both sides by \( \ln(4) \): \[ x = \frac{\ln(5)}{\ln(4)} \] Plug this into a calculator, and you'll find the approximate value for \( x \). Now, about logarithms: they’re not just for solving equations! Think of them as your math superhero when dealing with exponential growth in the real world, like population growth, finance, or even in physics with radioactive decay. Using logarithms can help you understand and better visualize these processes swiftly and efficiently. And if you're wondering about common mistakes, watch out for mixing up the bases while applying logarithm rules, especially when the bases differ (such as 4 and 5 in this case). Always ensure your calculator is set to the right mode—whether it's natural or base ten logarithm—to avoid confusion and errors in your final answer!