NMTC Inter Level Problem 5

Algebra Level 3

The number of solutions for ( l o g 10 x ) 2 = l o g 10 ( 100 x ) (log_{10}x)^2=log_{10}(100x) is

0 4 2 1

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2 solutions

Aditya Raut
Aug 27, 2014

log 10 100 x = log 10 100 + log 10 x \log_{10} 100x = \log_{10} 100+\log_{10} x

Let log 1 0 x = a \log_10 x =a

Thus a 2 = a + 2 a^2=a+2

This is a quadratic with roots a = 2 a=2 , a = 1 a=-1

Thus log 10 x = 1 or 2 x = 1 10 or 100 \log_{10} x=-1 \text{ or } 2 \implies x = \dfrac{1}{10} \text{ or } 100 , they're 2 \boxed{2} solutions.

( log 10 x ) 2 = log 10 ( 100 x ) = log 10 ( 100 ) + log 10 x = 2 + log 10 x ( log 10 x ) 2 log 10 x 2 = 0 (\log_{10} x)^2 =\log_{10}(100 x) =\log_{10} (100) +\log_{10} x= 2+\log_{10} x~~\implies\\ (\log_{10} x)^2- \log_{10} x-2=0
This is a quadratic equation in variable log x \log x . Hence two solutions.

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