Question

If \[f\left( x \right) = \dfrac{{{7^{1 + \ln x}}}}{{{x^{\ln 7}}}}\] then \[f\left( {2015} \right) = \]
A) 20
B) 7
C) 2015
D) 100

Answer 1

Hint:
We will find the value of the function at a point \[x\] by using the property of logarithm. First, we will separate the two terms in the numerator by using the properties of exponent. Then we will use logarithmic property to solve the function. We will then substitute 2015 in place of \[x\] to find the required answer.

Complete step by step solution:
The function given to us is \[f\left( x \right) = \dfrac{{{7^{1 + \ln x}}}}{{{x^{\ln 7}}}}\].
We know that by the property of exponent if the power of the exponent is added than we can separate them as two exponents with same base and the two different powers
So, we can separate the term in numerator as,
\[ \Rightarrow f\left( x \right) = \dfrac{{{7^1} \times {7^{\ln x}}}}{{{x^{\ln 7}}}}\]
Now using the property of logarithm where \[{a^{\ln b}} = {b^{\ln a}}\] , we will change the denominator. Therefore, we get
\[ \Rightarrow f\left( x \right) = \dfrac{{{7^1} \times {7^{\ln x}}}}{{{7^{\ln x}}}}\]
Cancelling the same term from numerator and denominator, we get
\[ \Rightarrow f\left( x \right) = 7\]
As we can see that the function has a constant value at every point \[x\].
So at \[x = 2015\]
\[f\left( {2015} \right) = 7\]

Hence, option (B) is correct.

Note:
While solving an exponent question having logarithm power, first we try to separate the terms by using exponent properties and then we try to cancel out the common term by using logarithm property. A logarithm is a power to which a number should be raised in order to get another number. An exponent symbolizes how many times a number is multiplied to itself. Exponents with the same base when multiplied there power is added and when they are divided there power is subtracted. In logarithm the logarithm of two variable products is equal to the sum of the logarithm of each variable separately.