Functions 06

Algebra Level 4

Find the number of real values of x x satisfying,
log ⁡ x 3 = 2 x − 3. \large \log_x 3 = 2 x - 3.

2 1 3 0

Raj Rajput by RAJ RAJPUT , 25, India

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2 solutions

Raj Rajput
Nov 9, 2015

13 Helpful 0 Interesting 0 Brilliant 0 Confused

How to get the answer without using graph

Praful Jain - 5 years, 7 months ago

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Refer the solution by Otto Bretscher sir!

Deepak Kumar - 5 years, 6 months ago

Nice problem and solution! However could you explain how to get the answer without graph?

Rohit Udaiwal - 5 years, 7 months ago

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Refer the solution by Otto Bretscher sir!

Deepak Kumar - 5 years, 6 months ago
Otto Bretscher
Nov 10, 2015

The function log ⁡ x 3 \log_{x}3 is decreasing on the interval ( 1 , ∞ ) (1,\infty) , with a range of ( 0 , ∞ ) (0,\infty) , so that it must intersect the increasing graph of 2 x − 3 2x-3 once, by the intermediate value theorem. Likewise, log ⁡ x 3 \log_{x}3 is decreasing on ( 0 , 1 ) (0,1) , with a range of ( − ∞ , 0 ) (-\infty,0) , again, with one point of intersection.

11 Helpful 1 Interesting 1 Brilliant 0 Confused

Elegant!!Knowing the fact that y=2x-3 is linear increasing helps find the answer without graph.

Deepak Kumar - 5 years, 6 months ago

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