Question

Write each of the following expressions as a single logarithm.1)log(2)+log(12)2)2log(4)+4log(3)

Answer 1

To solve both questions, we need to use the following two properties of the logarithm function.

$\begin{gathered} \log _b(x\cdot y)=\log _b(x)+\log _b(y) \\ \text{and} \\ \log _b(x^y)=y\log _b(x) \end{gathered}$

In our case,

$\begin{gathered} \log (2)+\log (12)=\log (2\cdot12)=\log (24) \\ \Rightarrow\log (2)+\log (12)=\log (24) \end{gathered}$

The answer to part 1) is log(24).

On the other hand,

$\begin{gathered} 2\log (4)=\log (4^2)=\log (16) \\ 4\log (3)=\log (3^4)=\log (81) \\ \Rightarrow2\log (4)+4\log (3)=\log (16)+\log (81)=\log (16\cdot81)=\log (1296) \\ \Rightarrow2\log (4)+4\log (3)=\log (1296) \end{gathered}$

The answer to part 2) is log(1296)