Question

How do you show\[{{x}^{\log y}}={{y}^{\log x}}\]?

Answer 1

Hint: In the given question, we have been asked to prove that the LHS of an expression is equal to the RHS of the expression and it is given that\[{{x}^{\log y}}={{y}^{\log x}}\]. In order to prove this given logarithmic expression, first we will need to take log function on both the sides of the expression or an equation. Later by using the property of logarithm which states that\[\log {{\left( x \right)}^{a}}=a\log x\], we will simplify the given logarithmic expression. Then due to commutative property of multiplication which states that in multiplication, you can multiply the given expression in any order, the final product will remain same such that; \[a\times b=b\times a\], the LHS will be equal to the RHS of the equation. Therefore, in this way we will prove that\[{{x}^{\log y}}={{y}^{\log x}}\].

Complete step-by-step solution:
We have given that,
\[\Rightarrow {{x}^{\log y}}={{y}^{\log x}}\]
Taking the log both the sides of the function, we will obtain
\[\Rightarrow \log {{x}^{\log y}}=\log {{y}^{\log x}}\]
Using the property of logarithm which states that \[\log {{\left( x \right)}^{a}}=a\log x\]
Applying this property in the above expression, we will get
\[\Rightarrow \left( \log y \right)\left( \log x \right)=\left( \log x \right)\left( \log y \right)\]
As we know that,
The commutative property of multiplication, which states that
In multiplication, you can multiply the given expression in any order. The final product will remain the same.
Such that, \[a\times b=b\times a\],
Applying this property in the above expression,
\[\Rightarrow \left( \log y \right)\left( \log x \right)=\left( \log x \right)\left( \log y \right)\]
Therefore,
\[{{x}^{\log y}}={{y}^{\log x}}\]
Hence proved.

Therefore \[{{x}^{\log y}}={{y}^{\log x}}\]

Note: To solve these types of questions, we used the basic formulas of logarithm. Students should always require to keep in mind all the formulae for solving the question easily. Also while solving the question they need to keep in mind all the basic properties of mathematical operations such as associative property, commutative property etc.