How do you use the properties of logarithms to expand \[\ln \left( {\dfrac{{{x^4}\sqrt y }}{{{z^5}}}} \right)\] ?

Hint: In this question, we have to expand the given logarithmic expression to get the solution.
First, we need to use the quotient property of logarithm to simplify it. Then applying product property to make it simpler. After that, apply exponent property of logarithm, we will get the required solution.

Formula used:
Logarithm formula is defined as,
\[{\log _a}\left( {bc} \right) = {\log _a}b + {\log _a}c\]
\[{\log _a}\left( {\dfrac{b}{c}} \right) = {\log _a}b - {\log _a}c\]
\[{\log _a}\left( {{b^c}} \right) = c{\log _a}b\]

Complete step-by-step answer:
We need to expand \[\ln \left( {\dfrac{{{x^4}\sqrt y }}{{{z^5}}}} \right)\] using the properties of logarithms.
First, we will apply quotient property of logarithm to split apart the logarithm, we get,
\[\ln \left( {\dfrac{{{x^4}\sqrt y }}{{{z^5}}}} \right) = \ln \left( {{x^4}\sqrt y } \right) - \ln \left( {{z^5}} \right)\]
Now, we can write the square root as,
\[\ln \left( {{x^4}\sqrt y } \right) - \ln \left( {{z^5}} \right) = \ln \left( {{x^4}{y^{\dfrac{1}{2}}}} \right) - \ln \left( {{z^5}} \right)\]
 Now, apply product property of logarithm to split apart the first logarithm, we get,
\[\ln \left( {{x^4}{y^{\dfrac{1}{2}}}} \right) - \ln \left( {{z^5}} \right) = \ln \left( {{x^4}} \right) + \ln \left( {{y^{\dfrac{1}{2}}}} \right) - \ln \left( {{z^5}} \right)\]
Lastly using exponent property to rewrite the logarithm we get,
\[\ln \left( {{x^4}} \right) + \ln \left( {{y^{\dfrac{1}{2}}}} \right) - \ln \left( {{z^5}} \right) = 4\ln x + \dfrac{1}{2}\ln y - 5\ln z\]

Hence, after expanding \[\ln \left( {\dfrac{{{x^4}\sqrt y }}{{{z^5}}}} \right)\] using the properties of logarithms we get, \[\ln \left( {{x^4}} \right) + \ln \left( {{y^{\dfrac{1}{2}}}} \right) - \ln \left( {{z^5}} \right) = 4\ln x + \dfrac{1}{2}\ln y - 5\ln z\].

Note:
Logarithm:
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x.
The logarithm of x to base b is denoted as \[{\log _b}x\] .
Properties of logarithm:
Exponent property: This property says that the log of a power is the exponent times the logarithm of the base of the power.
\[{\log _a}\left( {{b^c}} \right) = c{\log _a}b\]
Product property: This property says that the logarithm of a product is the sum of the logs of the factors.
\[{\log _a}\left( {bc} \right) = {\log _a}b + {\log _a}c\]
Quotient property: This property says that the log of a quotient is the difference of the logs of the dividend and the divisor.
\[{\log _a}\left( {\dfrac{b}{c}} \right) = {\log _a}b - {\log _a}c\]
These properties apply for any values of b, c and a for which each logarithm is defined, which is,
\[b,c > 0\] and \[0 < b \ne 1\] .