If log 10 2- log 102 x -1 and log 102 x -3 be three consecutive term of an AP- thenA- x-0B- x-1 C- x-log 2 5D- None of these

The correct option is C Given log_10 2, log_10(2^x 1)and log_10(2^x + 3) are in AP. ⇒ 2log_10(2^x 1) =log_10 2+log_10 (2^x+3) ⇒ (2^x 1)^2=2(2^x+3) ⇒ 2^2x 2 ...

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Byju's Answer

Standard XII

Mathematics

Fundamental Laws of Logarithms

If log 10 2, ...

Question

If $l o g_{10} 2, l o g_{10} (2^{x} - 1) a n d l o g_{10} (2^{x} + 3)$ be three consecutive term of an AP, then

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Solution

The correct option is C
Given $l o g_{10} 2, l o g_{10} (2^{x} - 1) a n d l o g_{10} (2^{x} + 3)$ are in AP.
$\Rightarrow 2 l o g_{10} (2^{x} - 1) = l o g_{10} 2 + l o g_{10} (2^{x} + 3) \Rightarrow (2^{x} - 1)^{2} = 2 (2^{x} + 3) \Rightarrow 2^{2 x} - 2. 2^{x} + 1 = 2. 2^{x} + 6 \Rightarrow 2^{2 x} - 4. 2^{x} - 5 = 0 \Rightarrow (2^{x} - 5) (2^{x} - 1) = 0 \Rightarrow 2^{x} = 5 \Rightarrow x = l o g_{2} 5 \Rightarrow 2^{x} = - 1 (n o t p o s s i b l e)$

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If log10 2,log10(2x−1)and log10(2x+3) be three consecutive term of an AP, then

The correct option is C Given log10 2,log10(2x−1)and log10(2x+3) are in AP. ⇒2log10(2x−1)=log102+log10(2x+3)⇒(2x−1)2=2(2x+3)⇒22x−2.2x+1=2.2x+6⇒22x−4.2x−5=0⇒(2x−5)(2x−1)=0⇒2x=5⇒x=log25⇒2x=−1 (not possible)

If $l o g_{10} 2, l o g_{10} (2^{x} - 1) a n d l o g_{10} (2^{x} + 3)$ be three consecutive term of an AP, then