SOLUTION: If 6^[1-(2/x)] = 3 Find 2^x

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Question 1209382: If 6^[1-(2/x)] = 3
Find 2^x
Found 3 solutions by math_tutor2020, Edwin McCravy, greenestamps:
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Let's solve for x.
6^[ 1-(2/x) ] = 3
log( 6^[ 1-(2/x) ] ) = log(3)
(1-2/x)*log(6) = log(3)
1-2/x = log(3)/log(6)
(x-2)/x = log(3)/log(6)
(x-2)*log(6) = log(3)*x
x*log(6)-2*log(6) = log(3)*x
x*log(6)-log(3)*x = 2*log(6)
x*(log(6)-log(3)) = 2*log(6)
x*log(6/3) = 2*log(6)
x*log(2) = log(6^2)
x = log(36)/log(2)
x = log2(36)
Then,
2^x = 2^[ log2(36) ] = 36

Or after you determine x you could say,
y = 2^x
log(y) = log(2^x)
log(y) = x*log(2)
log(y) = (log(36)/log(2))*log(2)
log(y) = log(36)
y = 36
2^x = 36

Answer by Edwin McCravy(20064) About Me  (Show Source):
You can put this solution on YOUR website!
matrix%282%2C1%2C%22%22%2C6%5E%281-2%2Fx%29+=+3%29
matrix%282%2C1%2C%22%22%2C6%5E1%2A6%5E%28-2%2Fx%29+=+3%29
matrix%282%2C1%2C%22%22%2C2%2A6%5E%28-2%2Fx%29+=+1%29
matrix%282%2C1%2C%22%22%2C6%5E%28-2%2Fx%29+=+1%2F2%29
matrix%282%2C1%2C%22%22%2C6%5E%28-2%2Fx%29+=+2%5E%28-1%29%29
Raise both sides to the -1 power
matrix%282%2C1%2C%22%22%2C6%5E%282%2Fx%29+=+2%29
matrix%282%2C1%2C%22%22%2C2%5E%282%2Fx%29%2A3%5E%282%2Fx%29+=+2%29
matrix%282%2C1%2C%22%22%2C3%5E%282%2Fx%29+=+2%5E%281-2%2Fx%29%29
Take log base 2 of both sides:
expr%282%2Fx%29log%282%2C%283%29%29=1-2%2Fx
expr%282%2Fx%29%28log%282%2C%283%29%29%2B1%29=1
2%28log%282%2C%283%29%29%2B1%29=x
2%2Alog%282%2C%283%29%29%2B2=x
Raise 2 to the power of each side
2%5E%282%2Alog%282%2C%283%29%29%2B2%29=2%5Ex
2%5E%282%2Alog%282%2C%283%29%29%29%2A2%5E2=2%5Ex
matrix%282%2C1%2C%22%22%2C2%5E%28log%282%2C%283%5E2%29%29%29%2A4=2%5Ex%29
2%5E%28log%282%2C%289%29%29%29%2A4=2%5Ex
9%2A4=2%5Ex
36=2%5Ex
Answer: 36

Edwin

Answer by greenestamps(13209) About Me  (Show Source):
You can put this solution on YOUR website!


A solution without using logarithms....

6%5E%281-2%2Fx%29=3

Raise each side of the equation to the power x

6%5E%28x-2%29=3%5Ex

Rewrite the left side as the product of powers of 2 and 3

%282%5E%28x-2%29%29%283%5E%28x-2%29%29=3%5Ex
2%5E%28x-2%29=%283%5Ex%29%2F%283%5E%28x-2%29%29=3%5E2

Multiply both side by 2^2

2%5Ex=%283%5E2%29%282%5E2%29=36

ANSWER: 2%5Ex=36

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A similar solution without using logarithms....

6%5E%281-2%2Fx%29=3

%286%29%286%5E%28-2%2Fx%29%29=3

6%5E%28-2%2Fx%29=1%2F2

Raise each side to the power x

6%5E%28-2%29=%281%2F2%29%5Ex

Take reciprocals of each side

6%5E2=2%5Ex

ANSWER: 2%5Ex=6%5E2=36