Question

Identify the function based on the given information.

1. Domain of the function is $R^{+}$

2. $F(x_1.x_2)=F\left(x_1\right)+F\left(x_2\right)$

3. $F\left(\frac{x_1}{x_2}\right)=F\left(x_1\right)-F\left(x_2\right)$

4. $F\left(1\right)=0$

5. $F\left(2^m\right)$ is proportional to $m$.

6. Range of function is $R$

• A. Logarithm Function
• B. Exponential Function
• C. Absolute value function
• D. $F\left(x\right)=\sqrt{x}$

Answer 1

The correct option is A Logarithm Function

Based on given information

Domain of Function is $R^{+}or(0,\infty )$

From the given options, we can guess it can be logarithm function because domain for other functions is R or $[0,\infty )$

2.$F(x_1,x_2)=F\left(x_1\right)+F\left(x_2\right)$

It is the property of logarithm function

${\log }_{10}ab={\log }_{10}a+{\log }_{10}b$

So,it should be a logarithm function

Other statements also validate the points.

3.$\log \left(\frac{a}{b}\right)=\log a-\log b$

4.${\log }_{10}\left(1\right)=0$

5.$\log 2^m=m\log 2$

$\log 2^m$ is proportional to $m$

7. Range of logarithm Function is R or $(-\infty ,\infty )$

Graph of ${\log }_{m}x\text{when}m>1$

Graph of ${\log }_{m}x$ when $0<m<1$