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. $\ln(6)$ 

      Step 1(i):

      $\ln6=\ln(2·3)$

      Step 2(i):

      $\ln6=\ln2+\ln3$

      Answer (i):

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

   2. $\ln(12)$ 

      Hint (ii):

      See part (a).

   3. $\ln(16)$ 

      Hint (iii):

      See part (a).

   4. $\ln(\frac{1}{3})$ 

      Step 1(iv):

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

      Step 2(iv):

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