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Math Help - Help with solving exponential equation

  1. #1
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    Help with solving exponential equation

    Solve the following equation for 'm':

    (1/5)^m * (1/4)^18 = 1/(2*(10)^35)

    I don't remember how to solve this, and I need help. A step by step solution would be very helpful.

    Thanks!
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  2. #2
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    \displaystyle \left(\frac{1}{5}\right)^m  = \frac{1^m}{5^m} =  \frac{1}{5^m}

    so

    \displaystyle \frac{1}{5^m}\times \left(\frac{1}{4}\right)^{18} =\frac{1}{2\times (10)^{35}}


    \displaystyle \frac{1}{5^m}\times \frac{1}{4^{18}} =\frac{1}{2\times (10)^{35}}

    \displaystyle 5^m = \frac{2\times (10)^{35}}{4^{18}}\implies m= \dots
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  3. #3
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    So here's the thing: I have been out of school for several years, and I'm thinking about going back to get a Master's Degree. I was taking a practice GMAT test and this was one of the questions. I can do this problem with a calculator, but I am supposed to be able to do this by hand for the test. I understand everything you've done so far, but I still don't know how to do this by hand. I am pretty sure once I see this worked out that it will come back to me.
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  4. #4
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    I have done it all bar the last step.

    It is taking log_5 of both sides.
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  5. #5
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    10= 2(5) and 4= 2(2) so you can write everything on the right as powers of 2 and 5.

    Do it! You'll like it!
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  6. #6
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    \displaystyle 5^m = \frac{2 \cdot 10^{35}}{4^{18}}

    \displaystyle 5^m = \frac{2 \cdot 10^{35}}{2^{36}}

    \displaystyle 5^m = \frac{10^{35}}{2^{35}} = 5^{35}
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  7. #7
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    Thanks so much for the help. You guys are the best!
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  8. #8
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    1/5)^m = 1/5^m, and (1/4)^18 = 1/4^18 (since 1 raised to any power = 1)
    Additionally, 4^18 = (2^2)^18 = 2^36.

    Since both sides of the equation have 1 in the their numerators, to set them = we have to set the demoninators equal to each other.

    The denominator of the left side is now = 5^m*2^36

    The denominator of the right side = 2(10)^35, which can be rewritten as (2)(2*5)^35, or (2)(2^35)(5^35).
    Since (2)(2^35)= 2^36, 2^36 can be cancelled from each side of the equation. We are now left with 5^m = 5^35. m = 35, and the correct answer is D




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