Question

How do you write the logarithmic expressions as a single logarithm ln( x + 2 ) – 2ln(x) ?

Answer 1

Hint: Solving logarithmic expressions are quite easy. There are some properties which will be used to solve such expressions.
Power rule: \[a\log x=\log {{x}^{a}}\]
Quotient rule: \[\log x-\log y=\log \left( \dfrac{x}{y} \right)\]
The properties used for log and ln are the same. So, there is no need to change the question in the form of log.

Complete step by step answer:
Now, solve this logarithmic expression.
First, write down the logarithmic expression which is given in question:
$\Rightarrow $ ln( x + 2 ) – 2ln(x)
As we know that the main difference between log and ln is: log is defined for base 10 and ln is defined for base e. So the properties used for log and ln are the same.
By using the power rule:
$a\log x=\log {{x}^{a}}$
The expression will be:
$\Rightarrow \ln (x+2)-\ln ({{x}^{2}})$
Here, we can see that 2 will become the power of ‘x’ with the help of the power rule which is used.
Now, further we will be using the quotient rule:
$\Rightarrow \log x-\log y=\log \left( \dfrac{x}{y} \right)$
$\Rightarrow \ln (x+2)-\ln ({{x}^{2}})$
As we can see in the above expression that there is ‘-‘ sign, so it is following the quotient rule stated above.

So, on further solving, we will get:
$\Rightarrow \ln \left( \dfrac{x+2}{{{x}^{2}}} \right)$
So, finally we got the expression as a single logarithm.

Note: You need to check each term of the expression if any property is applicable there or not. Then only further apply the property on the whole expression, otherwise first you need to solve each term to its end and then apply property on the whole expression.