Question

$a^{{\log }_{c}b}=b^{{\log }_{c}a}\{a,b,c>0andc\ne 1\}$

• A. True
• B. False

Answer 1

The correct option is A

True

Let's assume $a^{{\log }_{c}b}=b^{{\log }_{c}a}$

Taking log on both sides

${\log }_{c}b.{\log }_{10}a={\log }_{c}a.{\log }_{10}b$ ...........(1)

We know ${\log }_{a}b=\frac{{\log }_{10}b}{{\log }_{10}a}$

From equation 1

$\frac{{\log }_{10}b}{{\log }_{10}c}\times {\log }_{10}a=\frac{{\log }_{10}a}{{\log }_{10}c}\times {\log }_{10}b$

$\frac{{\log }_{10}a\times {\log }_{10}b}{{\log }_{10}c}=\frac{{\log }_{10}a\times {\log }_{10}b}{{\log }_{10}c}$

Here, we observe that both LHS $\&$ RHS are same. So, our assumption is correct and given statement is also true.