Find b − d when log a ⁡ b = 3 2 and log c...

Kade Reese

Kade Reese

Answered

2022-07-14

Find b d when log a b = 3 2 and log c d = 5 4
a , b , c are three natural numbers such that log a b = 3 2 and log c d = 5 4 . Given: a c = 9
Find b d

Answer & Explanation

yermarvg

yermarvg

Expert

2022-07-15Added 19 answers

Hint:
log a b = 3 2 b = a 3 2 a  is a perfect square,  a = α 2 b = α 3
log c d = 5 4 d = c 5 4 = ( a 9 ) 5 4 a 9  is a perfect 4-th power α 2 9 = β 4 α 2 β 4 = ( α + β 2 ) ( α β 2 ) = 9
Since α + β 2 > 0 the other factor of 9 ( α β 2 ) must be > 0
{ α + β 2 = 1 , α β 2 = 9 α + β 2 = 3 , α β 2 = 3 α + β 2 = 9 , α β 2 = 1
{ Impossible since  α + β 2 α β 2 α = 3 , β = 0 α = 5 , β = 2
{ a = 3 2 = 9 , b = 3 3 = 27 , c = 9 9 = 0 , d = 0 5 4 = 0 , impossible since  d > 0 a = 5 2 = 25 , b = 5 3 = 125 , c = 25 9 = 16 , d = 16 5 4 = 2 5 = 32
The only solution is
a = 25 b = 125 c = 16 d = 32
b d = 125 32 = 93

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?