# Logarithm Rules – Exercise 941

## Exercise

$$\frac{1}{2}+\log_9 {\sqrt{18}}-\log_9 {\frac{1}{\sqrt{2}}}+\log_9 4.5=$$

$$\frac{1}{2}+\log_9 {\sqrt{18}}-\log_9 {\frac{1}{\sqrt{2}}}+\log_9 4.5=2$$

## Solution

Using Logarithm Rules we get:

$$\frac{1}{2}+\log_9 {\sqrt{18}}-\log_9 {\frac{1}{\sqrt{2}}}+\log_9 4.5=$$

$$=\frac{1}{2}+\log_9 {\sqrt{9}\sqrt{2}}-\log_9 {\sqrt{2}^{-1}}+\log_9 4.5=$$

$$=\frac{1}{2}+\log_9 {3\sqrt{2}}-\log_9 {\sqrt{2}^{-1}}+\log_9 4.5=$$

$$=\frac{1}{2}+\log_9 3 +\log_9 \sqrt{2}-\log_9 {\sqrt{2}^{-1}}+\log_9 4.5=$$

$$=\frac{1}{2}+\log_9 3 +\log_9 \sqrt{2}-(-1)\log_9 {\sqrt{2}}+\log_9 4.5=$$

$$=\frac{1}{2}+\log_9 3 +\log_9 \sqrt{2}+\log_9 {\sqrt{2}}+\log_9 4.5=$$

$$=\frac{1}{2}+\frac{\log_3 3}{\log_3 9} +\log_9 \sqrt{2}+\log_9 {\sqrt{2}}+\log_9 4.5=$$

$$=\frac{1}{2}+\frac{\log_3 3}{\log_3 3^2} +\log_9 \sqrt{2}+\log_9 {\sqrt{2}}+\log_9 4.5=$$

$$=\frac{1}{2}+\frac{\log_3 3}{2\log_3 3} +\log_9 \sqrt{2}+\log_9 {\sqrt{2}}+\log_9 4.5=$$

$$=\frac{1}{2}+\frac{1}{2} +\log_9 \sqrt{2}+\log_9 {\sqrt{2}}+\log_9 4.5=$$

$$=\frac{1}{2}+\frac{1}{2} +\log_9 {(\sqrt{2}\sqrt{2})}+\log_9 4.5=$$

$$=\frac{1}{2}+\frac{1}{2} +\log_9 {2}+\log_9 4.5=$$

$$=1 +\log_9 {2}+\log_9 4.5=$$

$$=1 +\log_9 {2}+\log_9 \frac{9}{2}=$$

$$=1 +\log_9 {2}+\log_9 9 - \log_9 2=$$

$$=1 +\log_9 {2}+1 - \log_9 2=$$

$$=1 +1 =2$$

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