# CAT Practice : Exponents and Logarithms

2 can be expressed as a power of 3, 3 can be expressed as a power of 5. Do not live in the world that has only rational numbers; it is irrational to do that.

## Logartihm - Algebra

Q.4: If log1227 = a, log916 = b, find log8108.
1. ${2(a + 3) \over 3b}$
2. ${2(a + 3) \over 3a}$
3. ${2(b + 3) \over 3a}$
4. ${2(b + 3) \over 3b}$

Choice D. ${2(b + 3) \over 3b}$

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## Detailed Solution

log8108 = log8(4 × 27)
log8108 = log84 + log827
log84 = ${2 \over 3}$
log827 = ${log_{16} 27 \over log_{16} 8}$
log827 = ${log_{16} 27 \over 3/4}$
log827 = ${4 \over 3}$ log1627
log827 = ${{4 \over 3} {log_9{27} \over log_9{16}}}$
log827 = ${{4 \over 3} {3/2 \over log_9{16}}}$
log827 = 2 × log169
log916 = b
log169 = ${1 \over b}$
log827 = ${2 \over b}$
log8108 = ${{2 \over 3} {+} {2 \over b}}$
= $2{({1 \over 3} {+} {1 \over b})}$
= ${2(b + 3) \over 3b}$. Answer choice (D)

Correct Answer: ${2(b + 3) \over 3b}$

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