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Question: How do you find the constants \( a \) and \( b \) ? \( f\left( x \right) = a\ln \left( {bx} \righ...

How do you find the constants aa and bb ?
f(x)=aln⁡(bx)f\left( x \right) = a\ln \left( {bx} \right)
where f(e)f\left( e \right) and f′(2)=2f'\left( 2 \right) = 2

Explanation

Solution

Hint : We have been given a logarithmic function in terms of variable xx . aa and bb are constants in the function whose values are not known. We have been given two conditions for the function which are the values of the function and its derivative at certain points. We can find the values of these constants by forming two equations using these conditions and solving them.

Complete step by step solution:
We have been given the function f(x)=aln⁡(bx)f\left( x \right) = a\ln \left( {bx} \right) .
Here aa and bb are constants whose values are not known and we have to find the values.
We have also been given two conditions over the function and its derivative,
f(e)=2f\left( e \right) = 2 and f′(2)=2f'\left( 2 \right) = 2 .
Using f(e)=2f\left( e \right) = 2 in the given function, we can write,
f(e)=2 ⇒aln⁡(be)=2 ⇒a(ln⁡b+ln⁡e)=2 ⇒a(1+ln⁡b)=2   f\left( e \right) = 2 \\\ \Rightarrow a\ln \left( {be} \right) = 2 \\\ \Rightarrow a\left( {\ln b + \ln e} \right) = 2 \\\ \Rightarrow a\left( {1 + \ln b} \right) = 2 \;
We got our first equation. We will find the second equation using the second condition.
We can find the first derivative of the given function as,
f′(x)=d(aln⁡(bx))dx=a.1bx.b=axf'\left( x \right) = \dfrac{{d\left( {a\ln \left( {bx} \right)} \right)}}{{dx}} = a.\dfrac{1}{{bx}}.b = \dfrac{a}{x}
Thus, from f′(2)=2f'\left( 2 \right) = 2 we have,
f′(2)=2 ⇒a2=2⇒a=4   f'\left( 2 \right) = 2 \\\ \Rightarrow \dfrac{a}{2} = 2 \Rightarrow a = 4 \;
We got the value as a=4a = 4 .
We can use the value of aa to find the value of bb ,
a(1+ln⁡b)=2 ⇒4(1+ln⁡b)=2 ⇒1+ln⁡b=12 ⇒ln⁡b=−12 ⇒b=e−12=1e   a\left( {1 + \ln b} \right) = 2 \\\ \Rightarrow 4\left( {1 + \ln b} \right) = 2 \\\ \Rightarrow 1 + \ln b = \dfrac{1}{2} \\\ \Rightarrow \ln b = - \dfrac{1}{2} \\\ \Rightarrow b = {e^{ - \dfrac{1}{2}}} = \dfrac{1}{{\sqrt e }} \;
Hence, we got the values as a=4a = 4 and b=1eb = \dfrac{1}{{\sqrt e }} .
Thus the given function is f(x)=4ln⁡(xe)f\left( x \right) = 4\ln \left( {\dfrac{x}{{\sqrt e }}} \right)
So, the correct answer is “ f(x)=4ln⁡(xe)f\left( x \right) = 4\ln \left( {\dfrac{x}{{\sqrt e }}} \right) ”.

Note : We solved the problem of finding the constants by forming equations using the given conditions and then solving the equations. When we have to find two constants we require two equations and therefore two conditions must be given in the question. We can check the result by comparing the value of the function at a point given in the condition in the question.