Solve for x: $5 ^ { - x / 3 } = 0.0052$ . Round to 3 decimal places.
A)9.803
B)15.777
C)20.606
D)-20.606
E)-3.268

Question

Quizplus · Accepted Answer

To solve $5^{-x/3} = 0.0052$, first express 0.0052 in terms of a power of 5. Since $5^{-3} = 0.008$, and 0.0052 is close to this value, we need to find an exact representation. Convert 0.0052 to a power of 10, which is $5.2 \times 10^{-3}$. Recognizing that $10^{-3} = (5 \times 2)^{-3} = 5^{-3} \times 2^{-3}$, we adjust to keep the base 5, understanding that the exact match might not be straightforward without logarithms. Using logarithms, we solve as follows:$$5^{-x/3} = 0.0052$$Taking the log base 10 of both sides (or natural log, as the base of log doesn't matter in this context for solving for x):$$-\frac{x}{3} \log(5) = \log(0.0052)$$Solving for x gives:$$x = 3 \frac{\log(0.0052)}{\log(5)}$$Using a calculator:$$x \approx 9.803$$This calculation directly solves for x, rounding to three decimal places as requested.

Solve for X 5−x/3=0.00525 ^ { - x / 3 } = 0.00525−x/3=0.0052

Solve for X $5 ^ { - x / 3 } = 0.0052$