The repeated-product proof below covers positive integer \(r\). For any real \(r\), write \(b=a^u\), where&nbsp;\(u=\log_a b\); then&nbsp;\(b^r=a^{ur}\), which proves the same law.
All real logarithms below have positive arguments and a positive base different from 1. In the power law, the argument before raising it to a power is positive. Statements labelled as common mistakes describe rules that are not identities; equality can still occur for particular allowed values.
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:
The formula \({{\left( {{\log }_{a}}x \right)}^{3}}= 3{{\log }_{a}}x\) is not an identity.
But \({{\left( {{\log }_{a}}x \right)}^{3}}=\left( {{\log }_{a}}x \right)\left( {{\log }_{a}}x \right)\left( {{\log }_{a}}x \right)\)

The repeated-product proof below covers positive integer \(r\). For any real \(r\), write \(b=a^u\), where \(u=\log_a b\); then \(b^r=a^{ur}\), which proves the same law.

All real logarithms below have positive arguments and a positive base different from 1. In the power law, the argument before raising it to a power is positive. Statements labelled as common mistakes describe rules that are not identities; equality can still occur for particular allowed values.

\({{\log }_{a}}{{b}^{r}}=r{{\log }_{a}}b\)

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

The formula \({{\left( {{\log }_{a}}x \right)}^{3}}= 3{{\log }_{a}}x\) is not an identity.

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

Video Solution:

Solution:

Solution locked