Solving an exponential equation by using logarithms Solve for \( x \). \[ 11^{-2 x}=15^{-x-2} \]

To solve the equation \( 11^{-2x} = 15^{-x-2} \), start by applying logarithms to both sides to bring down the exponents: \[ \log(11^{-2x}) = \log(15^{-x-2}). \] Using the power rule of logarithms, this can be rewritten as: \[ -2x \log(11) = (-x-2) \log(15). \] Next, distribute the logarithm on the right side: \[ -2x \log(11) = -x \log(15) - 2 \log(15). \] Now, rearrange the equation to group the terms involving \( x \): \[ -2x \log(11) + x \log(15) = -2 \log(15). \] Factor out \( x \) from the left side: \[ x(-2 \log(11) + \log(15)) = -2 \log(15). \] Finally, solve for \( x \): \[ x = \frac{-2 \log(15)}{-2 \log(11) + \log(15)}. \] Simplifying gives you your solution for \( x \). Now, let’s make this more engaging! Understanding how logarithms work can be quite electrifying, just like the difference between a jam session and a symphony orchestra! Logarithms allow us to unravel the mysteries of exponents, but they also help in fields like finance—think compound interest—as well as in science for measuring pH and sound intensity; they truly rock in the real world! So how can you avoid common pitfalls when dealing with log equations? Always remember to check if the bases are the same before diving in! Also, ensure that your logarithm inputs are positive; you can't take the log of a negative number or zero! A little extra care will save you from calculation chaos!