Chapter 5.8, Problem 29AYU

To determine

Use properties of logarithm to write each logarithm in terms of $a$ and $b$ .

Answer to Problem 29AYU

  $\boxed{\ln 6=a+b}$

Explanation of Solution

Given information:

Suppose that $\ln 2=a$ and $\ln 3=b$ . Use properties of logarithm to write each logarithm in terms of $a$ and $b$ .

  $\ln 6$

Calculation:

Consider the given expression,

  $\ln 6$

  $6=2\times 3$

Apply logarithm property $M=Nthen{\log }_{a}M={\log }_{a}N$ (base can $10$ or $a$ )

  $\begin{array}{l}\ln 6=\ln \left(2\cdot 3\right)\\ \ln 2+\ln 3\end{array}$

Substitute $a$ for $2$ and $b$ for $3$ .

  $\begin{array}{l}\ln 2+\ln 3=a+b\\ \ln 6=a+b\end{array}$

Hence, $\boxed{\ln 6=a+b}$