Question

use natural logarithms to evaluate the logarithm to the nearest thousandth.\\(\log_{2}50\\)\\(\log_{2}50 = \square\\) (round to the nearest thousandth as needed.)

Explanation:

Step1: Apply Change of Base Formula

The change of base formula for logarithms is $\log_b a = \frac{\ln a}{\ln b}$. For $\log_2 50$, we have $a = 50$ and $b = 2$, so $\log_2 50 = \frac{\ln 50}{\ln 2}$.

Step2: Calculate Natural Logarithms

Using a calculator, $\ln 50 \approx 3.91202$ and $\ln 2 \approx 0.69315$.

Step3: Divide the Values

Divide $\ln 50$ by $\ln 2$: $\frac{3.91202}{0.69315} \approx 5.64386$.

Step4: Round to Nearest Thousandth

Rounding $5.64386$ to the nearest thousandth gives $5.644$.

Answer:

$5.644$