Question

Simplify, in each case state which Logarithmic Law you are using:a. 210912 3 + 4l09122b. logg25 + logg10 - 3logg5C. 4log32 - log34 – 2log3V3 – log312-

Answer 1

`\begin{gathered} \text{Given } \\ 2\log _(12)3+4\log _(12)2 \\ \end{gathered}`

We want to simplify the above expression

Applying Change of base Rule Law of logaritm:

`\begin{gathered} \\ \\ n\log _aM=\log _aM^n \end{gathered}`

So also,

`\begin{gathered} 2\log _(12)3=\log _(12)3^2 \\ 4\log _(12)2=\log _(12)2^4 \end{gathered}`
`\begin{gathered} 2\log _(12)3+4\log _(12)2\text{ =}\log _(12)3^2\text{ + }\log _(12)2^4 \\ \\ =\log _(12)9\text{ + }\log _(12)16 \\ \end{gathered}`

Applying Quotient Rule Law of logarithm

`\begin{gathered} \log _aM\text{ + }\log _aN\text{ = }\log _a(MN) \\ \end{gathered}`

So also,

`\begin{gathered} \log _(12)9\text{ + }\log _(12)16=\text{ }\log _(12)(9\text{ x 16)} \\ =\log _(12)144 \\ =\log _(12)12^2 \\ =2\log _(12)12 \\ (\text{Recall that Log}_aa=1) \\ Thus,\text{ }2\log _(12)12\text{ = 2(1)} \\ =2 \\ \end{gathered}`
`Hence,\text{ }2\log _(12)3+4\log _(12)2\text{ = 2}`