# CAT Practice : Exponents and Logarithms

When we change the base, Log questions become far tougher.

## Basic identities of Logarithm

Q.7: $Log_{3}x + Log_{x}3 = \frac{17}{4}.$Find x.
1. $3^{4}$
2. $3^{\frac{1}{8}}$
3. $3^{\frac{1}{4}}$
4. $3^{\frac{1}{3}}$

Choice C. $3^{\frac{1}{4}}$

## Detailed Solution

$Log_{3}x + Log_{x}3 = \frac{17}{4}.$
Let y = $Log_{3}x$
We know that $Log_{x}3 = \frac{1}{Log_{3}x}.$
Hence $Log_{x}3 = \frac{1}{y}.$
Thus the equation can be written as $y + \frac{1}{y} = \frac{17}{4}$
$4y^{2} + 4 = 17y$
$4y^{2} + 4 - 17y = 0$
Solving the above equation we get $y = 4 or \frac{1}{4}$
If y = 4
$Log_{3}x = 4$
Then $x = 3^{4}$
If $y = \frac{1}{4}$
$Log_{3}x = \frac{1}{4}$
Then $x = 3^{\frac{1}{4}}$

Correct Answer: $3^{\frac{1}{4}}$

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