Question

How do you solve $\ln ({{e}^{7x}})=15$ ?

Answer 1

Hint: It is quite easy to solve logarithmic expressions. Here in this question, we have to find the value of ‘x’. There are some properties that will be used to solve such expressions.
Logarithmic function is denoted as: ${{\log }_{b}}x=n$ or ${{b}^{n}}=x$
Power rule: \[a\log x=\log {{x}^{a}}\]
\[\Rightarrow ln\left( e \right)\text{ }=\text{ }1\] (logarithm of base is 1)
The properties used for log and \[\ln \] are the same. So, there is no need to change the question in form of log. Here, ‘e’ is the natural base in the logarithm equation.

Complete step by step answer:
Now, let’s solve the question.
As we know the logarithm is nothing but the power to which a number must be raised to get some other values and is the most convenient way to express large numbers.
There are only two types of logarithm: common logarithm and natural logarithm.
A common logarithm is denoted as log base 10 or simply log. Whereas natural log is denoted by the natural base i.e. ‘e’ and represented as ln or loge.
There are some rules to solve logarithms. They are as follows:
Product rule: ${{\log }_{b}}(mn)={{\log }_{b}}m+{{\log }_{b}}n$
Quotient rule: ${{\log }_{b}}\left( \dfrac{m}{n} \right)={{\log }_{b}}m-{{\log }_{b}}n$
Power rule: ${{\log }_{b}}({{m}^{n}})=n{{\log }_{b}}m$
Now, write the logarithm given in the question.
$\Rightarrow \ln ({{e}^{7x}})=15$
As we know Power rule: ${{\log }_{b}}({{m}^{n}})=n{{\log }_{b}}m$. By applying this rule, we get:
$\Rightarrow 7x\ln (e)=15$
We also know that the value of ln(e) = 1. So, now the logarithm will be:
$\Rightarrow $7x = 15
Now, we have to find the value of ‘x’ by keeping the ‘x’ alone. We will get:
$\therefore x=\dfrac{15}{7}$ = 2.1 So this is our final answer.

Note:
There is no need to substitute the value of ‘e’ as 2.71828. The easiest way is to apply rules and then find the answer but before that, you should know all the rules of logarithms. 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.