Question 169886
problem:
{{{ 2.79^(x-1) - 4.377^x = 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^(x-1) - 4.377^x = 0}}}
add 4.377^x to both sides of the equation:
{{{2.79^(x-1) = 4.377^x}}}
raise both sides of the equation to the {{{(1/(x-1))}}} exponent.
your equation becomes:
{{{(2.79^(x-1))^(1/(x-1)) = (4.377^x)^(1/(x-1))}}}
simplify this and it becomes:
{{{2.79^1 = 4.377^(x/(x-1))}}}
which is the same as:
{{{2.79 = 4.377^(x/(x-1))}}}
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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^x}}} if and only if {{{log(a,y) = x}}}
second formula:
{{{log(a,y) = log(10,y) / log(10,a)}}}
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using the first formula, your formula of:
{{{2.79 = 4.377^(x/(x-1))}}}
becomes:
{{{log(4.377,2.79) = x/(x-1)}}}
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using the second formula,
{{{log(4.377,2.79) = x/(x-1)}}}
becomes:
{{{(log(10,2.79) / log(10,4.377)) = x/(x-1)}}}
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this becomes:
{{{.69497895 = x/(x-1)}}}
let the letter "a" represent ".69497895" for now to make simplifying easier to show.
equation becomes:
{{{a = x/(x-1)}}}
multiply both sides of equation by (x-1)
{{{a*(x-1) = x}}}
simplify:
{{{a*x - a = x}}}
add a to both sides of equation and subtract x from both sides of equation:
{{{a*x - x = a}}}
factor out the x:
{{{x*(a-1) = a}}}
divide both sides by (a-1)
{{{x = a/(a-1)}}}
replace a with .69497895 and solve"
{{{x = .69497895/(1-.69497895)}}}
{{{x = .69497895/-.30502105}}}
{{{x= -2.278462259}}}
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plug that value of x into your original equation:
{{{2.79^(x-1) - 4.377^x = 0}}}
becomes:
{{{2.79^(-3.278462259) - 4.377^(-2.278462259) = 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.