(a) If \( \log _{x} 8=3 \), then \( x=\square \) (b) If \( \log _{x} 27=3 \), then \( x=\square \)

To solve \( \log_{x} 8 = 3 \), we can rewrite it in exponential form: \( x^3 = 8 \). This means \( x = 8^{1/3} \), giving us \( x = 2 \). For \( \log_{x} 27 = 3 \), we similarly convert it to exponential form: \( x^3 = 27 \). Hence, \( x = 27^{1/3} \), resulting in \( x = 3 \). So, we find \( x = 2 \) for part (a) and \( x = 3 \) for part (b). Find joys in exponential growth! Logs and exponents hold the key to understanding many real-life phenomena, from sound intensity levels to earthquake magnitudes. When tackling log equations, remember you’re essentially flipping the script on exponential relationships - think of it like converting a recipe into a grocery list! Common mistakes often include forgetting to convert back to exponential form or miscalculating the roots. Always check your work by plugging the value back into the logarithmic equation—it’s like verifying the pizza order before it's baked!