Question

question 10 of 10
which answer represents the domain of the logarithmic function given below?
$f(x) = \log_{7}x$

a. $x \geq 0$
b. $x < 0$
c. $x > 0$
d. all real numbers

Explanation:

Step1: Recall Logarithmic Domain Rule

The logarithmic function \( \log_b(x) \) (where \( b>0, b
eq1 \)) is defined only when the argument \( x \) is positive. This is because \( \log_b(x) = y \) is equivalent to \( b^y = x \), and an exponential function \( b^y \) (with \( b>0 \)) always outputs a positive value, so \( x \) must be positive for the equation to have a solution.

Step2: Apply Rule to \( F(x)=\log_7 x \)

For the function \( F(x) = \log_7 x \), the base \( 7>0 \) and \( 7
eq1 \), so we apply the domain rule for logarithms. The argument here is \( x \), so \( x \) must be greater than \( 0 \) (i.e., \( x > 0 \)).

Now we analyze the options:

• Option A: \( x\geq0 \) includes \( x = 0 \), but \( \log_7(0) \) is undefined (since \( 7^y = 0 \) has no solution for real \( y \)), so A is incorrect.
• Option B: \( x < 0 \) is invalid because the argument of a logarithm can't be negative (as exponential functions with positive base don't output negative numbers), so B is incorrect.
• Option C: \( x > 0 \) matches the domain rule for logarithms, so C is correct.
• Option D: "All real numbers" is incorrect because it includes negative numbers and zero, both of which make the logarithm undefined.

Answer:

C. \( x > 0 \)