Question

What are the domain and range of the logarithmic function f(x) = log7x? Use the inverse function to justify your answers.

Answer 1

The domain  is  x > 0  or (0,∞)

the range is  f(x) >= 0

the inverse of the function is  f-1(x)= (10^x ) / 7 

here we see that domain is >= 0  which is the range of f(x) and the range is > 0 whch equals the domain of f(x)

    

Answer 2

We have been given the function

$f(x)=\log 7x$

Domain is the set of x values for which the function is defined. We know that a logarithm function is defined for all values greater than zero.

Thus, for the domain , we have

$7x>0\\x>0$

The domain is given by $\left \{ x \in R, x>0 \right \}$

The range is the set of y values for which the function is defined. For this logarithm function, we can take any values. Hence, the range of the function is all real number.

Now let us find the inverse of the function.

$y=\log (7x)\\ \text{Swap x and y, we get}\\ x=log(7y)\\ \text{Now, we find the value of y}\\ 7y=10^x\\ y=\frac{1}{7}10^x\\ f^{-1}(x)=\frac{1}{7}10^x$

The domain of the inverse function is all real number and range is all values greater than zero.

We know that the domain of the function is range of the inverse function and the range of the function is domain of the inverse function.

Therefore, it justifies our solution.