### Home > INT3 > Chapter 10 > Lesson 10.3.3 > Problem10-185

10-185.

Consider that $\text{ln}\left(2\right) ≈ 0.69315$ and $\text{ln}\left(3\right) ≈ 1.0986$.

1. Why is $\ln(2)<1$ and $\ln(3)>1$? For what value of $x$ does $\ln(x)=1$?

2. Use the properties of logarithms to write each value below as an expression involving $\ln(2)$ and/or $\ln(3)$. Then use the given values for $\ln(2)$ and $\ln(3)$ to evaluate each of the expressions without a calculator.

1. $\ln(6)$

$\ln6=\ln(2·3)$

$\ln6=\ln2+\ln3$

$\ln6=(0.69315)+(1.0986)$
$\ln6=1.79175$

1. $\ln(12)$

See part (a).

1. $\ln(16)$

See part (a).

1. $\ln(\frac{1}{3})$

$\ln\frac{1}{3} = \ln\left(3^{-1}\right)$

$\ln\frac{1}{3} = -\ln3$