Question

use the aleks calculator to evaluate each expression. round your answers to the nearest thousandth. do not round any intermediate computations. \\(\log 32.5 = \square\\) \\(\ln \sqrt{3} = \square\\)

Explanation:

Step1: Evaluate $\log 32.5$

Using a calculator, we find the common logarithm (base 10) of 32.5.
$\log 32.5 \approx 1.511883$
Rounding to the nearest thousandth, we look at the fourth decimal place, which is 8. Since 8 ≥ 5, we round up the third decimal place.
So, $\log 32.5 \approx 1.512$

Step2: Evaluate $\ln \sqrt{3}$

First, we can rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$. Then, using the property of logarithms $\ln a^b = b\ln a$, we have:
$\ln \sqrt{3} = \ln 3^{\frac{1}{2}} = \frac{1}{2}\ln 3$
Using a calculator, $\ln 3 \approx 1.098612$
Then, $\frac{1}{2} \times 1.098612 = 0.549306$
Rounding to the nearest thousandth, we look at the fourth decimal place, which is 3. Since 3 < 5, we leave the third decimal place as is.
So, $\ln \sqrt{3} \approx 0.549$

Answer:

$\log 32.5 \approx \boxed{1.512}$, $\ln \sqrt{3} \approx \boxed{0.549}$