which of the following explains why $f(x) = \log_4 x$ does not have a y-intercept? check all that apply.

there is no power of 4 that is equal to 0.
there is no power of 4 that is equal to 1.
its inverse does not have any x-intercepts.
its inverse does not have any y-intercepts.

complete

the x-intercept of $f(x) = \log_4 x$ is

done

Question

which of the following explains why $f(x) = \log_4 x$ does not have a y-intercept? check all that apply.

there is no power of 4 that is equal to 0.
there is no power of 4 that is equal to 1.
its inverse does not have any x-intercepts.
its inverse does not have any y-intercepts.

complete

the x-intercept of $f(x) = \log_4 x$ is

done

Accepted Answer

{ "quality": "clear", "question_count": 2, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Logarithmic Functions", "Logarithmic and Exponential Inverses" ], "new_concepts": [ "Logarithmic Intercepts" ], "current_concepts": [ "Logarithmic Functions", "Logarithmic and Exponential Inverses", "Logarithmic Intercepts" ] } ## Analyze the y-intercept of the logarithmic function Using the *Logarithmic Functions* knowledge point $$ \begin{aligned} &y\text{-intercept occurs when } x = 0\ &f(0) = \log_4(0) \implies 4^y = 0\ &\text{Since } 4^y > 0 \text{ for all real } y, \text{ there is no power of 4 equal to 0} \end{aligned} $$ ## Analyze the inverse function relationship Using the *Logarithmic and Exponential Inverses* knowledge point $$ \begin{aligned} &f(x) = \log_4(x) \implies f^{-1}(x) = 4^x\ &\text{A } y\text{-intercept of } f(x) \text{ corresponds to an } x\text{-intercept of } f^{-1}(x)\ &\text{Since } 4^x = 0 \text{ has no real solution, the inverse } f^{-1}(x) \text{ has no } x\text{-intercepts} \end{aligned} $$ ## Determine the x-intercept of the logarithmic function Using the *Logarithmic Functions* knowledge point $$ \begin{aligned} &x\text{-intercept occurs when } f(x) = 0\ &\log_4(x) = 0 \implies x = 4^0 = 1 \end{aligned} $$ ## Question 1 * There is no power of 4 that is equal to 0 * Its inverse does not have any x-intercepts ## Question 2 The x-intercept of $f(x) = \log_4 x$ is 1 { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Logarithmic Functions" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze the y-intercept of the logarithmic function

Analyze the inverse function relationship

Determine the x-intercept of the logarithmic function

Answer:

Question 1

Question 2