Question

How do you use the property of algorithms to expand $\ln \left( {\dfrac{{{x^2} - 1}}{{{x^3}}}} \right)$?

Answer 1

Hint: We can expand $\ln \left( {\dfrac{{{x^2} - 1}}{{{x^3}}}} \right)$ by using the quotient property of algorithms. We simply use the quotient property and substitute the values in the formula from the expression given to us and will expand it further.

Complete step by step answer:
we have division inside the log. According to the quotient rule of the log, this can be split apart as subtraction outside the log, so:
Quotient Rule is given by: $\ln \left( {\dfrac{x}{y}} \right) = \ln \left( x \right) - \ln \left( y \right)$
Now, using the Quotient rule on the expression given to us we get:
$ \Rightarrow \ln \left( {\dfrac{{{x^2} - 1}}{{{x^3}}}} \right) = \ln \left( {{x^2} - 1} \right) - \ln \left( {{x^3}} \right)$
Always remember to take the time to check to see if any of the terms in your expansion can be simplified.
So, here we can see that to simplify the above expression further we have to use the Power Rule in $\ln \left( {{x^3}} \right)$but $\ln \left( {{x^2} - 1} \right)$remain unchanged because it cannot be expand further.
 Power Rule is given by: $\ln \left( {{x^a}} \right) = a \times \ln (x)$
Now, using the Power rule on the above expression we get:
$ \Rightarrow \ln \left( {\dfrac{{{x^2} - 1}}{{{x^3}}}} \right) = \ln \left( {{x^2} - 1} \right) - 3 \times \ln \left( x \right)$

Hence, the original expression $\ln \left( {\dfrac{{{x^2} - 1}}{{{x^3}}}} \right)$expanded fully as $\ln \left( {{x^2} - 1} \right) - 3\ln \left( x \right)$.

Note: Properties for expanding logarithm:
There are $5$ properties that are frequently used to expand logarithm.
Properties
 $1$: $\ln \left( 1 \right) = 0$…………………………… (Zero Exponent-Rule)
$2:$$\ln \left( a \right) = 1$
$3:$$\ln \left( {xy} \right) = \ln \left( x \right) + \ln \left( y \right)$………………………… (Product Rule)
$4:$$\ln \left( {\dfrac{x}{y}} \right) = \ln \left( x \right) - \ln \left( y \right)$…………………………. (Quotient Rule)
$5:$$\ln \left( {{x^a}} \right) = a \times \ln (x)$ …………………………………. (Power Rule)
Remember when we are expanding logarithms there is no specific order in which these properties must be applied but some guidelines are listed below:
$1.$ Rewrite any radicals using rational exponents (fraction).
$2.$Apply property $3$ or $4$ to rewrite the logarithm as addition and subtraction instead of multiplication and division.
$3$.Apply property $5$ to move the exponent out front of the algorithm.
$4.$ Apply property $1$ or $2$ to simplify the logarithms.