Question

How do you simplify \[{{e}^{3\ln (x)}}\]?

Answer 1

Hint: We will simplify the given expression using the properties of exponents. Firstly, we will assign the given expression to a variable, let’s say \[y={{e}^{3\ln (x)}}\]. Applying the properties of logarithm function in the above expression, that is, \[\ln ({{a}^{b}})=b\ln a\] and we get the simplified form of the given expression.

Complete step by step solution:
According to the given question, we have to simplify the expression based on the properties of exponents.
We will start by assigning a variable to the given expression and we get,
Let,
\[y={{e}^{3\ln (x)}}\]------(1)
Applying natural logarithm on both sides of the equality, we get,
\[\Rightarrow \ln y=3\ln (x)\]
Now, we will be applying a common property of logarithm function in the above expression. The property is, \[\ln ({{a}^{b}})=b\ln a\].
We get,
\[\Rightarrow \ln y=\ln ({{x}^{3}})\]-----(2)
We can write this obtained expression also by removing the logarithm function. So, we have,
\[\Rightarrow y={{x}^{3}}\]-----(3)
This is the simplified form of the expression.
If we compare the equation (3) with the equation (1), we get,
\[{{e}^{3\ln (x)}}={{x}^{3}}\]

Therefore, the simplified form of the expression \[{{e}^{3\ln (x)}}={{x}^{3}}\].

Note: We can carry out the above solution using another property of exponents, that is, \[{{e}^{{{\log }_{e}}}}=1\].
We will start by writing the given expression, we have,
\[{{e}^{3\ln (x)}}\]
\[\Rightarrow {{({{e}^{\ln x}})}^{3}}\]
We got the above step as we know that, \[{{a}^{(xy)}}={{({{a}^{x}})}^{y}}\]
\[\Rightarrow {{({{e}^{{{\log }_{e}}x}})}^{3}}\]
And we will now use the property of exponent that, exponents when raised to logarithm having base e equals to 1, that is, \[{{e}^{{{\log }_{e}}}}=1\] and also using the formula \[{{a}^{{{\log }_{a}}b}}=b\], we get,
\[\Rightarrow {{(x)}^{3}}\]
As \[{{e}^{{{\log }_{e}}x}}=x\], so we got the above expression.
\[\Rightarrow {{x}^{3}}\]
Therefore, the simplified form of the expression \[{{e}^{3\ln (x)}}={{x}^{3}}\].