If log 4 5- a and log 5 6- b- then log 3 2 equal toA- 1-2 a-1B- 2 b-1C- 2 a b-1D- 2 ab -1

The correct option is C 12ab 1log_45=a⇒ 4^a=5 log_56=b⇒ 5^b=6 ⇒6=(4^a)^b=2^2ab ⇒3=2^2ab2⇒3=2^2ab 1 Let log_32=x⇒3^x=2 ⋯(1) ⇒3^x=2^(2ab 1)x ⋯(2) From (1) an ...

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Byju's Answer

Standard XII

Mathematics

Fundamental Laws of Logarithms

If log 4 5= a...

Question

If ${log}_{4} 5 = a$ and ${log}_{5} 6 = b$ , then ${log}_{3} 2$ equal to

\frac{1}{2 a + 1}

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\frac{1}{2 b + 1}

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2 a b + 1

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\frac{1}{2 a b - 1}

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Solution

The correct option is D $\frac{1}{2 a b - 1}$
${log}_{4} 5 = a \Rightarrow 4^{a} = 5 {log}_{5} 6 = b \Rightarrow 5^{b} = 6 \Rightarrow 6 = (4^{a})^{b} = 2^{2 a b} \Rightarrow 3 = \frac{2^{2 a b}}{2} \Rightarrow 3 = 2^{2 a b - 1}$
$Let {log}_{3} 2 = x \Rightarrow 3^{x} = 2 \dots (1) \Rightarrow 3^{x} = 2^{(2 a b - 1) x} \dots (2)$
$From (1) and (2) (2 a b - 1) x = 1 x = \frac{1}{2 a b - 1}$

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If log45=a and log56=b, then log32 equal to

The correct option is D 12ab−1log45=a⇒4a=5log56=b⇒5b=6⇒6=(4a)b=22ab⇒3=22ab2⇒3=22ab−1 Let log32=x⇒3x=2 ⋯(1)⇒3x=2(2ab−1)x ⋯(2) From (1) and (2)(2ab−1)x=1x=12ab−1

If ${log}_{4} 5 = a$ and ${log}_{5} 6 = b$ , then ${log}_{3} 2$ equal to