Solve the given equation for \( x \). \( 3^{5 x-2}=35 \) \( x=\square \) Question Help: \( \square \) Video

To solve the equation \( 3^{5x - 2} = 35 \), you can take the logarithm of both sides. Using natural logarithms, you get: \[ \ln(3^{5x - 2}) = \ln(35) \] This can be simplified using the power rule of logarithms: \[ (5x - 2) \ln(3) = \ln(35) \] Now, isolate \( 5x - 2 \): \[ 5x - 2 = \frac{\ln(35)}{\ln(3)} \] Next, add 2 to both sides: \[ 5x = \frac{\ln(35)}{\ln(3)} + 2 \] Finally, divide by 5 to solve for \( x \): \[ x = \frac{1}{5} \left( \frac{\ln(35)}{\ln(3)} + 2 \right) \] With the numeric calculations, the approximate value of \( x \) will help in inserting into the square. For a fun trivia fact, logarithms were first introduced by mathematician John Napier in the early 17th century as a way to simplify complex calculations, making them hugely impactful in fields such as astronomy and navigation. To practically apply logarithms in everyday life, you can use them while computing things like pH levels in chemistry or sound intensity in decibels. Understanding logarithmic scales can be helpful in interpreting these real-world measurements!