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Math Help - Dealing with exponent selection.

  1. #1
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    Dealing with exponent selection.

    What would be the fastest way to deal with questions like these ?

    If n is an integer in 5n > 4,000,000 , what is the least possible value of n ?
    7,8,9,10 or 11 (Ans=10)
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  2. #2
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    Re: Dealing with exponent selection.

    Are you allowed to use a calculator? If so n>log_5 4000000.

    If not then log_5 4000000=log_5(5^6 \times 2^6 \times 4) =6+log_5(256)=9 \text{ point something}.
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  3. #3
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    Re: Dealing with exponent selection.

    I need it without a calculator. How did u get

    log5(56 x 26 x 4)
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  4. #4
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    Re: Dealing with exponent selection.

    Hello, MikeNoob!

    \text{If }n\text{ is an integer such that: }\:5^n \:<\:4,\!000,\!000
    \text{what is the least possible value of }n?

    Here is a very primitive approach . . .

    We have: . 5^n \:>\:4,\!000,\!000 \;=\;2^8\cdot5^6 \;=\;256\cdot5^6 \;>\;250\cdot5^6

    . . . . . . . . 5^n \;>\;2\cdot125\cdot5^6 \;=\;2\cdot5^3\cdot5^6 \;=\;2\cdot5^9 \;>\;5^9

    Hence: . 5^n \;>\;5^9 \quad\Rightarrow\quad n \;>\;9


    \text{Therefore, the least value of }n\text{  is }10.


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


    \text{Check: }\:\begin{Bmatrix}5^9 &=& 1,953,125 \\ 5^{10} &=& 9,765,625\end{Bmatrix}
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