Power Law
${{\log }_{a}}{{b}^{r}}=r{{\log }_{a}}b$
e.g. ${{\log }_{2}}{{5}^{x}}=x{{\log }_{2}}5$ , ${{\log }_{3}}\sqrt{2}=\frac{1}{2}{{\log }_{3}}2$
Important note:
${{\left( {{\log }_{a}}x \right)}^{3}}\ne 3{{\log }_{a}}x$
But ${{\left( {{\log }_{a}}x \right)}^{3}}=\left( {{\log }_{a}}x \right)\left( {{\log }_{a}}x \right)\left( {{\log }_{a}}x \right)$

e.g. ${{\log }_{2}}{{5}^{x}}=x{{\log }_{2}}5$ , ${{\log }_{3}}\sqrt{2}=\frac{1}{2}{{\log }_{3}}2$

${{\left( {{\log }_{a}}x \right)}^{3}}\ne 3{{\log }_{a}}x$

But ${{\left( {{\log }_{a}}x \right)}^{3}}=\left( {{\log }_{a}}x \right)\left( {{\log }_{a}}x \right)\left( {{\log }_{a}}x \right)$

Question 115739

Question 115739

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