Let f(x) be defined for all x 0 and be cont | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Given the functional equation $f(\frac{x}{y}) = f(x) - f(y)$.Setting $y = 1$,we get $f(x) = f(x) - f(1)$,which implies $f(1) = 0$.Setting $x = 1$,

Vedclass · Accepted Answer

$f(x) = \ln(x)$

Vedclass · Answer

(D) Given the functional equation $f(\frac{x}{y}) = f(x) - f(y)$.Setting $y = 1$,we get $f(x) = f(x) - f(1)$,which implies $f(1) = 0$.Setting $x = 1$,we get $f(\frac{1}{y}) = f(1) - f(y) = -f(y)$.For any $x, y > 0$,$f(xy) = f(\frac{x}{1/y}) = f(x) - f(1/y) = f(x) + f(y)$.This is the Cauchy functional equation for the logarithm function.Since $f(x)$ is continuous and satisfies $f(xy) = f(x) + f(y)$,the solution is of the form $f(x) = c \ln(x)$.Given $f(e) = 1$,we have $c \ln(e) = 1$,so $c(1) = 1$,which means $c = 1$.Thus,$f(x) = \ln(x)$.Therefore,option $D$ is correct.

Let $f(x)$ be defined for all $x > 0$ and be continuous. Let $f(x)$ satisfy $f(\frac{x}{y}) = f(x) - f(y)$ for all $x, y > 0$ and $f(e) = 1$. Then:

Similar Questions

Which of the following metallic hydroxides does not dissolve in a sodium hydroxide solution?

$ABC$ is a triangular plate of uniform thickness. The sides are in the ratio $3:4:5$ as shown in the figure. $I_{AB}, I_{BC}, I_{CA}$ are the moments of inertia of the plate about axes $AB, BC$ and $CA$ respectively. Which one of the following relations is correct?

Power delivered by the $ac$ source of the circuit becomes maximum when

The angle between the tangents drawn to the parabola $y^2=4x$ from the point $(1,4)$ is

What is microcosmic salt?

Vedclass Test Series

Exam Paper Generator

Online Exam Module