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

In the reaction,$C_2H_5OC_2H_5 + CO \xrightarrow[150^{\circ}C, 500 \text{ atm pressure}]{BF_3} X$. What is $X$?

If $l, m, n$ are in arithmetic progression,then the straight line $lx + my + n = 0$ will always pass through the point:

$A$ person is observing a bacteria through a compound microscope. For better observation and to improve its resolving power,he should:

Helium mixed with oxygen is used in the treatment of

Which one of the following pesticides is banned now-a-days?

Vedclass Test Series

Exam Paper Generator

Online Exam Module