# CAT Practice : Exponents and Logarithms

Basic Identities

## Basic identities of Logarithm

Q.8: $Log_{x}y + Log_{y}x^{2} = 3.$ Find $Log_{x}y^{3}.$
1. 4
2. 3
3. $3^{\frac{1}{2}}$
4. $3^{\frac{1}{16}}$

Choice B. 3

## Detailed Solution

$log_{x}y + log_{y}x^{2} = 3$
Let a = $log_{x}y$
$log_{x}3 = \frac{1}{log_{3}x}.$
$log_{y}x^{2} = 2log_{y}x$
We know that $log_{y}x = \frac{1}{log_{x}y}$
Hence form above $log_{y}x = \frac{1}{a}$
Now rewritting the equation $log_{x}y + log_{y}x^{2} = 3$
Using a we get $a + \frac{2}{a} = 3$
i.e., $a^{2} - 3a + 2 = 0$
Solving we get a = 2 or 1

If a = 2, Then $log_{x}y = 2$ and $log_{y}x^{3} = 3$
$log_{x}y = 3 * 2 = 6$

Or

If a = 1, Then $log_{x}y = 1$ and $log_{y}x^{3} = 3$
$log_{x}y = 3 * 1 = 3$

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