# CAT Practice : Exponents and Logarithms

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

## Logarithms - Different bases

Q.6: log5x = a (This should be read as log X to the base 5 equals a) log20x = b. What is logx10?
1. ${a + b \over 2ab}$
2. (a + b) * 2ab
3. ${2ab \over a + b}$
4. ${a + b \over 2}$

Choice A. ${a + b \over 2ab}$

## Detailed Solution

log5x = a

log20x = b

logx5 = ${1 \over a}$

logx20 = ${1 \over b}$

logx${100 \over 5} {=} {1 \over b}$

logx100 - logn5 = ${1 \over b}$

2logx10 - ${1 \over a}$ = ${1 \over b}$

logx10 = ${{1 \over 2} ({1 \over a} {+} {1 \over b})}$

logx10 = ${a + b \over 2ab}$

Correct Answer: ${a + b \over 2ab}$

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