Solve for $x : 5 ^ { - x / 2 } = 0.0052$. Round to 3 decimal places.
A) $6.535$
B) $10.518$
C) $13.737$
D) $- 13.737$
E) $- 3.268$

Question

Quizplus · Accepted Answer

To solve $5^{-x/2} = 0.0052$, first express 0.0052 in terms of a power of 5. Since $5^{-3} = 0.008$, and $0.0052$ is close to $0.008$, we can't directly express $0.0052$ as a power of 5. Instead, convert $0.0052$ to a power of 10, because it's given in decimal form: $0.0052 = 5.2 \times 10^{-3}$. Now, convert the equation to a logarithmic form or use the property that if $a^b = c$, then $b = \log_a(c)$. However, because we can't directly convert $0.0052$ to a base of 5, we use the natural logarithm for a more straightforward approach:$$5^{-x/2} = 0.0052$$Taking the natural logarithm of both sides:$$\ln(5^{-x/2}) = \ln(0.0052)$$Using the logarithm power rule, $\ln(a^b) = b\ln(a)$, we get:$$-\frac{x}{2} \ln(5) = \ln(0.0052)$$Solving for $x$:$$x = -2 \frac{\ln(0.0052)}{\ln(5)}$$Using a calculator:$$x \approx -2 \frac{\ln(0.0052)}{\ln(5)} \approx 6.535$$Therefore, the correct answer is $6.535$, rounded to three decimal places.

Solve For x:5−x/2=0.0052x : 5 ^ { - x / 2 } = 0.0052x:5−x/2=0.0052

Solve For $x : 5 ^ { - x / 2 } = 0.0052$