Functions
Logarithm equality and base determination
Grade Class 12
Question:
If the logarithms of two distinct positive numbers are equal to the corresponding numbers, then the base of the logarithm belongs to the interval
$(0,1)$
$(1,e)$
$(e, e^5)$
$(1, e^{1/e})$
Step-by-Step Solution
Key Concept: We need $\log_a b = b$ and $\log_a c = c$ for two distinct $b,c>0$. This means $a=b^{1/b}=c^{1/c}$. Use the graph of $y=x^{1/x}$.
Equation $a=x^{1/x}$ has two solutions for $x>1$ iff $a\in(1, e^{1/e})$.
Correct Answer: 4