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calculus online - Free exercises and solutions to help you succeed!

Logarithm Rules – Exercise 941

Exercise

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

Final Answer

\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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