Question

question 1 of 10
what is the solution to the equation below? round your answer to two decimal places.
\\(\log_{5} x = 2.1\\)
\\(\bigcirc\\) a. \\(x = 8.17\\)
\\(\bigcirc\\) b. \\(x = 10.50\\)
\\(\bigcirc\\) c. \\(x = 40.84\\)
\\(\bigcirc\\) d. \\(x = 29.37\\)

Explanation:

Step1: Recall log to exponential conversion

For $\log_b a = c$, we can rewrite it as $a = b^c$. So for $\log_5 x = 2.1$, we get $x = 5^{2.1}$.

Step2: Calculate $5^{2.1}$

Using a calculator, $5^{2.1} \approx 5^{2 + 0.1}=5^2\times5^{0.1}=25\times1.2589\approx31.47$? Wait, no, maybe my calculator approach is wrong. Wait, $5^{2.1}$: let's compute it properly. $5^{2.1}=\mathrm{e}^{2.1\ln 5}$. $\ln 5\approx1.6094$, so $2.1\times1.6094\approx3.38$, then $\mathrm{e}^{3.38}\approx29.37$. Ah, right. So $x = 5^{2.1}\approx29.37$.

Answer:

D. $x = 29.37$