Brennan Flores

2021-09-19

The expansion for $log\frac{{x}^{2}{y}^{3}}{\sqrt[3]{z}}$ using the properties of logarithms and to represent the answer without exponents.

mhalmantus

Given:
The expression $log\frac{{x}^{2}{y}^{3}}{\sqrt[3]{z}}$.
Formula used:
Properties of logarithm:
1) ${\mathrm{log}x}^{n}=n\mathrm{log}x$
2) $\mathrm{log}\left(xy\right)=\mathrm{log}x+\mathrm{log}y$
3) $\mathrm{log}\left(\frac{x}{y}\right)=\mathrm{log}x-\mathrm{log}y$
Calculation:
The given expression is $\mathrm{log}\frac{{x}^{2}{y}^{3}}{\sqrt[3]{z}}$.
Since the expression on which log is applied is a fraction so we will apply property (3).
Which gives $\mathrm{log}\frac{{x}^{2}{y}^{3}}{\sqrt[3]{z}}=\mathrm{log}\left({x}^{2}{y}^{3}\right)-\mathrm{log}\sqrt[3]{z}\cdots a\right)$
Now, in $\mathrm{log}\left({x}^{2}{y}^{3}\right)$, log is applied on product of two terms so we can apply property (2).
$⇒\mathrm{log}\left({x}^{2}{y}^{3}\right)=\mathrm{log}\left({x}^{2}\right)+\mathrm{log}\left({y}^{3}\right)$
Also, $\sqrt[3]{z}$ can be written as ${z}^{\frac{1}{3}}$.
So, putting these in equation a) we get,
$\mathrm{log}\left({x}^{2}{y}^{3}\right)=\mathrm{log}\left({x}^{2}\right)+\mathrm{log}\left({y}^{3}\right)-\mathrm{log}\left({z}^{\frac{1}{3}}\right)$
Now, on right hand side, all the expressions have exponents so we can apply property (1).
This gives:
$\mathrm{log}\left({x}^{2}{y}^{3}\right)=\mathrm{log}\left({x}^{2}\right)+\mathrm{log}\left({y}^{3}\right)-\mathrm{log}\left({z}^{\frac{1}{3}}$.
This is the required result in which there is no exponent.
Conclusion:
These type of expressions can be simplified easily by applying the properties of log. When log is applied to product of two or more terms then corresponding log terms get added and similarly, when log is applied to a fraction then corresponding log terms get subtracted as given in the properties. These properties are used to simplify the expression.
If any log term has exponent then the exponent gets multiplied with the log term as discussed in the first property. This property is used to get rid of exponents.

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