# Thread: Dealing with exponent selection.

1. ## 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)

2. ## Re: Dealing with exponent selection.

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

If not then $\displaystyle log_5 4000000=log_5(5^6 \times 2^6 \times 4) =6+log_5(256)=9 \text{ point something}$.

3. ## Re: Dealing with exponent selection.

I need it without a calculator. How did u get

log5(56 x 26 x 4)

4. ## Re: Dealing with exponent selection.

Hello, MikeNoob!

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

Here is a very primitive approach . . .

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

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

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

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

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