SOLUTION: If f(x)= 3(1/125)(^x-2)+12,then what is the value of x when f(x)=87?

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Question 126695: If f(x)= 3(1/125)(^x-2)+12,then what is the value of x when f(x)=87?
Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
Given:
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f%28x%29=+3%2A%281%2F125%29%5E%28x-2%29%2B12
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When f(x) = 87 this becomes:
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87+=+3%281%2F125%29%5E%28x-2%29%2B12
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Transpose this equation (just switch sides) so the unknown x is on the left side. This transposition
makes the equation become:
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3%281%2F125%29%5E%28x-2%29%2B12+=+87
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Get rid of the 12 on the left side by subtracting 12 from both sides to get:
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3%281%2F125%29%5E%28x-2%29+=+75
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Next, get rid of the multiplier 3 on the left side by dividing both sides by 3 to reduce
the equation to:
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%281%2F125%29%5E%28x-2%29+=+25
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On the left side of the equation, you can simplify it by applying the power rule of exponents
to both the numerator and the denominator of the fraction to get:
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1%5E%28x-2%29%2F125%5E%28x-2%29+=+25
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But for the numerator you can apply the rule that 1 raised to any power is just 1. When you
apply that rule, the equation becomes:
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1%2F125%5E%28x-2%29+=+25
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Get rid of the denominator by multiplying both sides of this equation by 125%5E%28x-2%29
and you have:
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1+=+125%5E%28x-2%29%2A%2825%29
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But 125 can be replaced by 5%5E3 and 25 can be replaced by 5%5E2. When you do that you get:
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1+=+%285%5E3%29%5E%28x-2%29%2A5%5E2
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Apply the power rule by raising the 5%5E3 to the x-2 power to get:
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1+=+5%5E%283%2A%28x-2%29%29%2A5%5E2
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Multiply out the exponent and you get:
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1+=+5%5E%283x-6%29%2A5%5E2
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When you multiply two common bases that are raised to an exponent, the rule is that you
add the exponents. So in this case you add 3x - 6 and 2 to get the exponent of 5 and the equation
then becomes:
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1+=+5%5E%283x+-+6+%2B+2%29
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Simplify the exponent by combining the -6 and the +2 and you have:
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1+=+5%5E%283x+-4%29
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Now here's a little tricky insight. If you raise any number to the zero power the answer
is 1. So if you raise 5 to the zero power it will equal the 1 on the left side of this equation.
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Therefore, we can say that the exponent 3x - 4 must equal zero. In equation form, this is:
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3x+-+4+=+0
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Get rid of the -4 on the left side by adding 4 to both sides of this equation to get:
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3x+=+4
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And solve for x by dividing both sides of this equation by 3 and you have the answer:
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x+=+4%2F3
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A little bit of work to get to this point. Hope this helps you to understand how to do the problem.
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