Solve the equation for x by using base 10 logarithms. 16*4^(2.5x)=70

Marcus Bass

Marcus Bass

Answered question

2022-09-17

solve the equation using logarithms (I think this is easy level)
Solve the equation for x by using base 10 logarithms.
16 4 2.5 x = 9
EDIT: I made a typo (somehow... I was very far off!!)
The correct equation is this:
16 4 2.5 x = 70
Can it be written like:
2.5 x log 10 ( 5 ) = 70   ?
Then get:
log 10 ( 5 ) = 70 2.5 x
The computer wants a largest value and smallest value, similar to an answer for a quadratic problem. I need to know how to get the 2 answers even if one ends up negative (I know the negative will be tossed out, but I still need to know how to get the answer).

Answer & Explanation

Matthias Calhoun

Matthias Calhoun

Beginner2022-09-18Added 11 answers

16 4 2.5 x = 70 2 4 ( 2 2 ) 2.5 x = 70 2 4 2 5 x = 70 2 4 + 5 x = 70 log 10 2 4 + 5 x = log 10 70 ( 4 + 5 x ) log 10 2 = log 10 70 4 + 5 x = log 10 70 log 10 2
Can you take it from here?
Addendum :
4 2 ( 2 2 ) 2.5 x = 70 4 2 ( 2 2 ) 2.5 x ( 70 ) 2 = 0 ( 4 2 2.5 x 70 ) ( 4 2 2.5 x + 70 ) = 0
It will yield two solutions like you want.
Harrison Mills

Harrison Mills

Beginner2022-09-19Added 2 answers

16 4 2.5 x = 16 ( 4 2.5 ) x = 16 ( 4 5 2 ) x = 16 32 x
So, 32 x = 9 16
Thus, x = log 32 ( 9 16 ) = log 10 ( 9 16 ) log 10 ( 32 )
For the updated equation
16 4 2.5 x = 16 ( 4 2.5 ) x = 16 ( 4 5 2 ) x = 16 32 x
So, 32 x = 30 16
Thus, x = log 32 ( 30 16 ) = log 10 ( 15 8 ) log 10 ( 32 )

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