SOLUTION: Please help me solve this equation: {{{ 2.79^(x-1) - 4.377^x = 0 }}}. Thank you!

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Question 169886: Please help me solve this equation: +2.79%5E%28x-1%29+-+4.377%5Ex+=+0+. Thank you!
Found 2 solutions by Alan3354, gonzo:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
+2.79%5E%28x-1%29+-+4.377%5Ex+=+0+
(x-1)*log(2.79) = x*log(4.377)
(x-1)*0.4456 = x*0.6412
0.4456x - 0.4456 = 0.6412x
-0.4456 = 0.19557x
x = -2.2785

Answer by gonzo(654) About Me  (Show Source):
You can put this solution on YOUR website!
problem:
+2.79%5E%28x-1%29+-+4.377%5Ex+=+0+
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not sure if this is the correct way to approach this, but i got a solution and it appears to be good.
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you start off with:
2.79%5E%28x-1%29+-+4.377%5Ex+=+0
add 4.377^x to both sides of the equation:
2.79%5E%28x-1%29+=+4.377%5Ex
raise both sides of the equation to the %281%2F%28x-1%29%29 exponent.
your equation becomes:
%282.79%5E%28x-1%29%29%5E%281%2F%28x-1%29%29+=+%284.377%5Ex%29%5E%281%2F%28x-1%29%29
simplify this and it becomes:
2.79%5E1+=+4.377%5E%28x%2F%28x-1%29%29
which is the same as:
2.79+=+4.377%5E%28x%2F%28x-1%29%29
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now that you have this in a standard exponential form, you should be able to solve it using logarithms.
you make use of two logarithmic formulas:
first formula:
y+=+a%5Ex if and only if log%28a%2Cy%29+=+x
second formula:
log%28a%2Cy%29+=+log%2810%2Cy%29+%2F+log%2810%2Ca%29
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using the first formula, your formula of:
2.79+=+4.377%5E%28x%2F%28x-1%29%29
becomes:
log%284.377%2C2.79%29+=+x%2F%28x-1%29
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using the second formula,
log%284.377%2C2.79%29+=+x%2F%28x-1%29
becomes:
%28log%2810%2C2.79%29+%2F+log%2810%2C4.377%29%29+=+x%2F%28x-1%29
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this becomes:
.69497895+=+x%2F%28x-1%29
let the letter "a" represent ".69497895" for now to make simplifying easier to show.
equation becomes:
a+=+x%2F%28x-1%29
multiply both sides of equation by (x-1)
a%2A%28x-1%29+=+x
simplify:
a%2Ax+-+a+=+x
add a to both sides of equation and subtract x from both sides of equation:
a%2Ax+-+x+=+a
factor out the x:
x%2A%28a-1%29+=+a
divide both sides by (a-1)
x+=+a%2F%28a-1%29
replace a with .69497895 and solve"
x+=+.69497895%2F%281-.69497895%29
x+=+.69497895%2F-.30502105
x=+-2.278462259
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plug that value of x into your original equation:
2.79%5E%28x-1%29+-+4.377%5Ex+=+0
becomes:
2.79%5E%28-3.278462259%29+-+4.377%5E%28-2.278462259%29+=+0
you will find the answer to be 0 proving the value of x is correct.
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i may have messed up the digits but the real values were stored in the calculator and are accurate.
the values should be:
.69497895 for a
-.30502105 for 1-a
-2.278462259 for x
any deviation from these numbers above is a typo.
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there might be an easier way to do this but i don't see it right now.