SOLUTION: can somebody help me solve this equation: {{{5^(x+2)=7^(x-2)}}}

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Question 169002This question is from textbook college algebra
: can somebody help me solve this equation: 5%5E%28x%2B2%29=7%5E%28x-2%29This question is from textbook college algebra

Answer by Electrified_Levi(103) About Me  (Show Source):
You can put this solution on YOUR website!
Hi, Hope I can help
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can somebody help me solve this equation: 5%5E%28x%2B2%29=7%5E%28x-2%29
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first we must take the log of both sides, log is the inverse function of an exponent, it is also the power of a base function
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The two sides will have a base of "10"
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log+%2810%2C+5%5E%28x%2B2%29%29=log+%28+10%2C7%5E%28x-2%29%29
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+log+%28+a%2C+b%5Ec+%29+=+%28c%29%28log%28a%2Cb%29%29+, if there is an exponent in a log such as (x+2), and (x-2), you take the exponents and multiply them by (
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+log+%2810%2C+base%29+), in this case +log+%28+10%2C5%29, and +log+%2810%2C7%29+
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%28x%2B2%29log+%2810%2C5%29=%28x-2%29log+%28+10%2C7%29,
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We can rearrange the two sides
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%28x%2B2%29log+%2810%2C5%29=%28x-2%29log+%28+10%2C7%29 = log+%2810%2C5%29%28x%2B2%29=log+%28+10%2C7%29%28x-2%29
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we can use the distribution to multiply the numbers out
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log+%2810%2C5%29%28x%2B2%29=log+%28+10%2C7%29%28x-2%29 = highlight%28log+%2810%2C5%29%29%28highlight%28x%29%2B2%29=highlight%28log+%28+10%2C7%29%29%28highlight%28x%29-2%29 = highlight%28log+%2810%2C5%29%29%28x%2Bhighlight%282%29%29=highlight%28log+%28+10%2C7%29%29%28x-highlight%282%29%29
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Remember the signs
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+x%28log+%28+10%2C5+%29%29+%2B+2log+%2810%2C5%29+=+x%28log+%28+10%2C7%29%29+-+2log+%28+10%2C7%29+,
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now we need to get all the "x"s on one side, we will move +x%28log+%28+10%2C7%29%29+, to the left side
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+x%28log+%28+10%2C5+%29%29+%2B+2log+%2810%2C5%29+=+x%28log+%28+10%2C7%29%29+-+2log+%28+10%2C7%29+ =
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+x%28log+%28+10%2C5+%29%29+%2B+2log+%2810%2C5%29+-+x%28log+%28+10%2C7%29%29+=+x%28log+%28+10%2C7%29%29+-+x%28log+%28+10%2C7%29%29+-+2log+%28+10%2C7%29+ =
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+x%28log+%28+10%2C5+%29%29+%2B+2log+%2810%2C5%29+-+x%28log+%28+10%2C7%29%29+=+-+2log+%28+10%2C7%29+
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Now we will move +2log+%2810%2C5%29+ to the right side
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+x%28log+%28+10%2C5+%29%29+%2B+2log+%2810%2C5%29+-+x%28log+%28+10%2C7%29%29+=+-+2log+%28+10%2C7%29+ =
.
+x%28log+%28+10%2C5+%29%29+%2B+2log+%2810%2C5%29+-+2log+%2810%2C5%29+-+x%28log+%28+10%2C7%29%29+=+-+2log+%28+10%2C7%29+-+2log+%2810%2C5%29+ =
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+x%28log+%28+10%2C5+%29%29+-+x%28log+%28+10%2C7%29%29+=+-+2log+%28+10%2C7%29+-+2log+%2810%2C5%29+
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We can take out the "x"s on the left side ( since they both have it in common
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+x%28log+%28+10%2C5+%29%29+-+x%28log+%28+10%2C7%29%29+=+-+2log+%28+10%2C7%29+-+2log+%2810%2C5%29+ = +x%28log+%28+10%2C5+%29+-+log+%28+10%2C7%29%29+=+-+2log+%28+10%2C7%29+-+2log+%2810%2C5%29+
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To find "x" you would divide both sides by +log+%28+10%2C5+%29+-+log+%28+10%2C7%29+
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+x%28log+%28+10%2C5+%29+-+log+%28+10%2C7%29%29+=+-+2log+%28+10%2C7%29+-+2log+%2810%2C5%29+ = +%28x%28log+%28+10%2C5+%29+-+log+%28+10%2C7%29%29%29%2F%28log+%28+10%2C5+%29+-+log+%28+10%2C7%29%29+=+%28-+2log+%28+10%2C7%29+-+2log+%2810%2C5%29%29%2F%28log+%28+10%2C5+%29+-+log+%28+10%2C7%29%29+ =
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+%28x+%28cross%28log+%28+10%2C5+%29+-+log+%28+10%2C7%29%29%29%29%2Fcross%28log+%28+10%2C5+%29+-+log+%28+10%2C7%29%29+=+%28-+2log+%28+10%2C7%29+-+2log+%2810%2C5%29%29%2F%28log+%28+10%2C5+%29+-+log+%28+10%2C7%29%29+ = +x+=+%28-+2log+%28+10%2C7%29+-+2log+%2810%2C5%29%29%2F%28log+%28+10%2C5+%29+-+log+%28+10%2C7%29%29+
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+x+=+%28-+2log+%28+10%2C7%29+-+2log+%2810%2C5%29%29%2F%28log+%28+10%2C5+%29+-+log+%28+10%2C7%29%29+ = +x+=+21.13308+ (rounded)(you have to solve using a calculator)
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You usually check by replacing "x" with the answer, but in this case, it would be hard since the answer never stops ( I checked it, and it is right )
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Hope I helped, Levi