Question

How do you use the properties of logarithms to expand $\ln z{\left( {z - 1} \right)^2}$ ?

Answer 1

Hint : A logarithm of a number with a base is equal to another number. A logarithm is just the opposite function of exponentiation. The natural logarithm is called the base e logarithm. To expand the given function, using the properties of logarithms we have; $\log x + \log y \Leftrightarrow \log \left( {xy} \right)$ and $\log {x^n} \Leftrightarrow n\log x$ , hence applying this we need to expand the given function.

Complete step-by-step answer :
Given,
$\ln z{\left( {z - 1} \right)^2}$ ,
Let, us use the following properties to expand the given function:
$\log x + \log y \Leftrightarrow \log \left( {xy} \right)$ and $\log {x^n} \Leftrightarrow n\log x$
Hence, applying this to the given function as:
$\Rightarrow \ln z{\left( {z - 1} \right)^2}$
We, know that $\log x + \log y \Leftrightarrow \log \left( {xy} \right)$ , hence we have:
$= \ln z + \ln z{\left( {z - 1} \right)^2}$
We, know that $\log {x^n} \Leftrightarrow n\log x$ , hence we get:
$= \ln z + 2\ln \left( {z - 1} \right)$
Therefore, we get
$\ln z{\left( {z - 1} \right)^2} = \ln z + 2\ln \left( {z - 1} \right)$
So, the correct answer is “$\ln z + 2\ln \left( {z - 1} \right)$ ”.

Note : A logarithm is defined as the power to which number must be raised to get some other values. A logarithm has various important properties that prove multiplication and division of logarithms can also be written in the form of logarithm of addition and subtraction.
Logarithmic Functions have some of the properties that allow you to simplify the logarithms when the input is in the form of product, quotient or the value taken to the power. Some of the rules and properties are:
Product Rule: Multiply two numbers with the same base, then add the exponents i.e., ${\log _b}MN = {\log _b}M + {\log _b}N$
Quotient Rule: Divide two numbers with the same base, subtract the exponents.
${\log _b}\dfrac{M}{N} = {\log _b}M - {\log _b}N$