Properties of Logs

log1. Evaluate each of the following expressions using the propeties of logs (and no calculator)

log_{3}\sqrt[3]{81}=log_{3}\left ( 81 \right )^{\frac{1}{3}}=log_{3}\left ( 3^{4} \right )^{\frac{1}{3}}=log_{3}3^{\frac{4}{3}}=\frac{4}{3}\cdot log_{3}3=\frac{4}{3}

ln\left ( lne^{e^{200}} \right )=ln\left ( e^{200}\cdot lne \right )=lne^{200}=200\cdot lne=200

2. Use the properties of logs to expand the following expressions.

log_{5}\sqrt[4]{x^{3}\left ( x^{2}+1 \right )}=log_{5}\left ( \sqrt[4]{x^{3}} \right )\left ( \sqrt[4]{x^{2}+1} \right )=log_{5}x^{\frac{3}{4}}\cdot \left ( x^{2}+1 \right )^{\frac{1}{4}}=log_{5}x^{\frac{3}{4}}+log_{5}\left ( x^{2}+1 \right )^{\frac{1}{4}}=\frac{3}{4}log_{5}x+\frac{1}{4}log_{5}(x^{2}+1)

log\sqrt{x\sqrt{y\sqrt{z}}}=log\sqrt{x\sqrt{y\cdot z^{\frac{1}{2}}}}=log\sqrt{x\cdot \left ( yz^{}\frac{1}{2} \right )^{\frac{1}{2}}}=log\sqrt{xy^{\frac{1}{2}}z^{\frac{1}{4}}}=logx^{\frac{1}{2}}y^{\frac{1}{4}}z^{\frac{1}{8}}=\frac{1}{2}logx+\frac{1}{4}logy+\frac{1}{8}logz

3. Use the properties of logs to condense the following expressions

4lnx-\frac{1}{3}ln\left ( x^{2}+1 \right )+2ln(x-1)=lnx^{4}-ln\left ( x^{2}+1 \right )^{\frac{1}{3}}+ln(x-1)^{2}=lnx^{4}+ln(x-1)^{2}-ln\sqrt[3]{x^{2}+1}=lnx^{4}(x-1)^{2}-ln\sqrt[3]{x^{2}+1}=ln\frac{x^{4}(x-1)^{2}}{\sqrt[3]{x^{2}+1}}

log(x^{2}-1)-ln(x-1)=\frac{ln(x^{2}-1)}{ln10}-ln(x-1)=\frac{1}{ln10}\cdot ln(x^{2}-1)-ln(x-1)=ln(x^{2}-1)^{\frac{1}{ln10}}-ln(x-1)=ln\frac{(x^{2}-1)^{\frac{1}{ln10}}}{x-1}

4. If  log_{7}x=Alog_{\frac{2}{3}}x, use the change of base formula to find the value of A.

log_{7}x=Alog_{\frac{2}{3}}x\Rightarrow A=\frac{log_{7}x}{log_{\frac{2}{3}}x}=\frac{\frac{logx}{log7}}{\frac{logx}{log\frac{2}{3}}}=\frac{log\frac{2}{3}}{log7}=log_{7}\frac{2}{3}

5. Simplify the following to a single log expression of the form logba:

\left (log_{7}3 \right )\left ( log_{2}5 \right )\left ( log_{5}7 \right )=\frac{log3}{log7}\cdot \frac{log5}{log2}\cdot \frac{log7}{log5}=\frac{log3}{log2}=log_{2}3