If 3(4^h)=4(2^k) and 9(8^h)=20(4^k),show that 2^h=4/5

Deja Bradshaw

Deja Bradshaw

Answered question

2022-10-18

If 3 ( 4 h ) = 4 ( 2 k ) and 9 ( 8 h ) = 20 ( 4 k ),show that 2 h = 4 5
I tried to substitute the equation 1 into equation 2 so that I can find the value of k or h, but it did not work as the base is not the same (I cannot compare to find the answer)
How can I find the answer?

Answer & Explanation

enracant60

enracant60

Beginner2022-10-19Added 10 answers

We can see that 2 h 4 h k = 20 9 . On the other hand 2 2 h k = 4 3 . So 4 h k 2 k h = 5 3 . This implies 2 3 h 3 k = 5 3 . Since 2 k = 3 4 2 2 h we have 2 3 k = 27 64 2 6 h . Therefore, 2 3 h 2 3 k = 64 27 2 3 h 2 6 h = 64 27 2 3 h = 5 3 . This implies ( 2 h ) 3 = 64 / 45
Martin Hart

Martin Hart

Beginner2022-10-20Added 3 answers

If you really want to use logarithms, then
log ( 3 ) + h log ( 4 ) = log ( 4 ) + k log ( 2 )
log ( 9 ) + h log ( 8 ) = log ( 20 ) + k log ( 4 )
Now, taking into account 9 = 3 2 , 8 = 2 3 , 4 = 2 2 , the expressions write
log ( 3 ) + 2 h log ( 2 ) = 2 log ( 2 ) + k log ( 2 )
2 log ( 3 ) + 3 h log ( 2 ) = log ( 20 ) + 2 k log ( 2 )
and you have two linear equations in h,k easy to solve (for example, multiply the first by 2 and subtract from the second to get h).
I am sure that you can take from here.

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