Question:
The range of the function $f(x)=\logₑ(3x²-4x+5)$ is ________.
The range of the function $f(x)=\logₑ(3x²-4x+5)$ is ________.
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The range of the function $f(x)=\loge(3x-4x+5)$ is ____.
Updated On: Apr 15, 2026
- $(-\infty, \log_{e}\frac{11}{3}]$
- $[\log_{e}\frac{11}{3}, \infty)$
- $[-\log_{e}\frac{11}{3}, \log_{e}\frac{11}{3}]$
- None of the above
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The Correct Option is B
Solution and Explanation
Step 1: Concept
Find the range of the quadratic expression $3x^2 - 4x + 5$ first.
Step 2: Quadratic Analysis
The minimum value of a quadratic $ax^2 + bx + c$ with $a>0$ is $\frac{4ac - b^2}{4a}$. Minimum value $= \frac{4(3)(5) - (-4)^2}{4(3)} = \frac{60 - 16}{12} = \frac{44}{12} = \frac{11}{3}$.
Step 3: Logarithmic Range
Since the minimum value of the inner expression is $11/3$ and it increases to infinity, the range of the log function is $[\log_{e}\frac{11}{3}, \infty)$.
Final Answer: (B)
Find the range of the quadratic expression $3x^2 - 4x + 5$ first.
Step 2: Quadratic Analysis
The minimum value of a quadratic $ax^2 + bx + c$ with $a>0$ is $\frac{4ac - b^2}{4a}$. Minimum value $= \frac{4(3)(5) - (-4)^2}{4(3)} = \frac{60 - 16}{12} = \frac{44}{12} = \frac{11}{3}$.
Step 3: Logarithmic Range
Since the minimum value of the inner expression is $11/3$ and it increases to infinity, the range of the log function is $[\log_{e}\frac{11}{3}, \infty)$.
Final Answer: (B)
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