Solve the given equation for \( x \). \( 2^{\frac{x}{2}}=5 \) \( x= \) Basic Funcs Trig

To solve the equation \( 2^{\frac{x}{2}} = 5 \), we'll take the logarithm of both sides. Applying the logarithm (let's use natural log, \( \ln \)) gives us: \[ \ln(2^{\frac{x}{2}}) = \ln(5) \] Using the property of logarithms that allows us to bring down the exponent, we have: \[ \frac{x}{2} \ln(2) = \ln(5) \] Now, multiplying both sides by 2 gives us: \[ x \ln(2) = 2 \ln(5) \] Finally, dividing both sides by \( \ln(2) \) yields: \[ x = \frac{2 \ln(5)}{\ln(2)} \] And that's your solution! For a fun fact, did you know that logarithms were invented in the early 17th century by John Napier? They were revolutionary in simplifying complex calculations, particularly in astronomy and navigation. So next time you encounter logs, remember you're delving into a mathematical tradition that dates back over 400 years! In practical terms, logarithms are like superheroes for solving exponential equations. They make it super easy to isolate variables and find solutions, especially in fields like physics, engineering, and finance. So, embracing logs is your secret weapon for tackling real-world problems!