Logarithm / exponential equation, not sure what to make of this, (simple) Solve for a : (

Leland Morrow

Leland Morrow

Answered question

2022-06-20

Logarithm / exponential equation, not sure what to make of this, (simple)
Solve for a : ( 2 log a x ) ( 3 log x 2 4 ) = 3
No idea how to approach this problem other than moving the 2 and the 3 into an exponent..

Answer & Explanation

pheniankang

pheniankang

Beginner2022-06-21Added 22 answers

Use change of base: log x y = log n y log n x
( 2 log a x ) ( 3 log x 2 4 ) = ( log a x 2 ) ( log x 2 4 3 ) = ln x 2 ln a ln 4 3 ln x 2 = ln 4 3 ln a = log a 4 3 = 3 a 3 = 4 3 a = 4
preityk7t

preityk7t

Beginner2022-06-22Added 6 answers

This answer is full of seemingly useless derivations, but would help you in the future problems. There are three interesting results of the base-change property:
First property: log a b = 1 log b a
Proof. log a b = log b log a and 1 log b a = ( log b a ) 1 = ( log a log b ) 1 = log b log a
Second property: log a m b n = n m log a b
Proof. As you may be aware of, log x n = n log x. Hence,
log a m b n = n log a m b = n 1 log b a m = n 1 m log b a = n m 1 log b a = n m log a b
Third property: a log x b = b log x a
Proof. a log a b log a x = ( a log a b ) 1 log a x = b 1 log a x = b log x a
As for this problem,
( 2 log a x ) ( 3 log x 2 4 ) = 3
log a x 2 3 log x 2 4 = 3
3 log x 2 4 log x 2 a = 3
log x 2 4 log x 2 a = 1
log a 4 = 1
a 1 = 4
a = 4

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