Dealing with exponent selection.

**MikeNoob** · July 19th 2012, 12:31 AM

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)

**a tutor** · July 19th 2012, 01:28 AM

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

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

**MikeNoob** · July 19th 2012, 03:54 PM

I need it without a calculator. How did u get

log₅(5⁶ x 2⁶ x 4)

**Soroban** · July 19th 2012, 05:23 PM

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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