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

Solution

$\begin{gathered} \text{Given } \\ 2\log _{12}3+4\log _{12}2 \\ \end{gathered}$

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}$