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Equations with variables in the exponents

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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by theCEO » Wed Jun 20, 2012 9:31 pm
5^21 * 4^11 = 2 * 10^n
5(5^20) * 4(4^10) = 2 * 10^n
(5^20) * (4^10) = 0.1 * 10^n
(5^10) * (5^10)* (4^10) = 0.1 * 10^n
(5*5*4)^10 = 0.1 * 10^n
100^10 = 0.1 * 10^n
10^20 = 10^-1 * 10^n
10^21 = 10^n
n = 21
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by theCEO » Wed Jun 20, 2012 9:43 pm
2^x - 2^x-2 = 3(2^13)
2^x - (2^x * 2^-2) = 3(2^13)
2^x(1-2^-2) = 3(2^13)
2^x(1-0.25) = 3(2^13)
2^x(3/4) = 3(2^13)
2^x = 3(2^13) * (4/3)
2^x = 4*(2^13)
2^x = 2^2*(2^13)
2^x = 2^15
x = 15
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by Mike@Magoosh » Thu Jun 21, 2012 4:10 pm
Kim9876Zey wrote:I think I am missing some of the rules in solving these
Dear Kim9876Zey

You might find this blog article a helpful summary of the laws of exponents.
https://magoosh.com/gmat/2012/exponent-p ... -the-gmat/

Here's a somewhat related practice question:
https://gmat.magoosh.com/admin/questions/913
When you submit your answer, the following page will have the complete video solution.

Let me know if you have any further questions.

Mike :)
Magoosh GMAT Instructor
https://gmat.magoosh.com/
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by jcnasia » Fri Jun 22, 2012 7:06 am
You can also solve this problem by taking the log of both sides.
log(5^21 * 4^11) = log(2 * 10^n)

21 * log 5 + 11 * log 4 = log 2 + n * log 10
21 * log 5 + 11 * 2 * log 2 - log 2 = n
21 * log 5 + 21 * log 2 = n
21 * log(5*2) = n
21 = n
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