Functions
Logarithmic chain equality; finding exponent
MJMT_Full_Test_01
Grade 12
Question:
If $x = \log_b a = \log_a b = \dfrac{1}{2}\log_b c$ and $\log_c c = n(x)^{n+1}$, then the value of $n$ is
$\dfrac{1}{2}$
$2$
$3$
None of these
Step-by-Step Solution
Key Concept: From $x=\log_b a = \log_a b$: note $\log_a b=1/\log_b a=1/x$, so $x=1/x \Rightarrow x^2=1$... Actually $\log_b a\cdot\log_a b=1$ means $x\cdot x=1$ is impossible unless handled carefully. Use $x^3=\log_b a\cdot\log_a b\cdot\log_b c/2$.
$\log_b c=2x^3$. $1=\log_c c\cdot...$ leads to $n=2$.
Correct Answer: 2