If $n$ is an integer and $5^n > 4,000,000.$ What is the least possible value of $n$? (answer: $10$)
How could I find the value of $n$ without using a calculator ?
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Sign up to join this communityIf $n$ is an integer and $5^n > 4,000,000.$ What is the least possible value of $n$? (answer: $10$)
How could I find the value of $n$ without using a calculator ?
\begin{eqnarray} & 5^n &>& 4.000.000\\ \Leftrightarrow & 5^n &>& 5^6 \cdot 2^8 \\ \Leftrightarrow & 5^{n-6} &>& 256.\\ \end{eqnarray} Then, $n=10$.
Divide 4000000 by 5, without a calculator, getting 800000. Divide again; 160000. Again; 32000. Then 6400, then 1280, then 256, then 51 (rounding), then 10, then 2. So $2\times5^9$ is about 4000000, so $5^{10}$ exceeds 4000000.
5^(n-308) > 2^312*3*13
or so and then need to estimate log(2)/log(5) part to enough precision that I can multiply by a three digit number and know I'm not off by 1 or introduce approximations 624 ~ 625. I can sort of estimate values (e.g., Newton's method) to get within about ~1 (I find N=444 instead of 445), but would be very hard to prove.
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– dr jimbob
Jul 31 '12 at 20:13
$4,000,000 = 2^2 \times 10^6 = 2^8 \times 5^6$, so you want $5^{n-6} > 2^8 = 256$. Well, $5^3 = 125\ldots$.
By logarithm rules: $$5^{n}>4\cdot10^{6}\iff n>\log_{5}2^{2}2^{6}5^{6}=\log_{5}2^{8}+\log_{5}5^{6}=\log_{5}2^{8}+6=\log_{5}256+6$$
Since these are relatively small numbers I assume it is ok to write : $5^{3}=125$ thus clearly $3<\log_{5}256<4$ hence the minimal $n$ that satisfies this inequality is $4+6=10$
I dunno, this is a tough one, especially without a calculator.
Here is the Python program I used to figure this one out:
for n in range(1,11):
print "5^%s-4,000,000 = %s" % (n, pow(5,n)-4000000)
Here is the output:
5^1-4,000,000 = -3999995.0
5^2-4,000,000 = -3999975.0
5^3-4,000,000 = -3999875.0
5^4-4,000,000 = -3999375.0
5^5-4,000,000 = -3996875.0
5^6-4,000,000 = -3984375.0
5^7-4,000,000 = -3921875.0
5^8-4,000,000 = -3609375.0
5^9-4,000,000 = -2046875.0
5^10-4,000,000 = 5765625.0
It looks like $n=10$ is the answer.
import math; math.ceil(math.log(4000000,5))
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– Alex L
Jul 31 '12 at 8:42
It helps if you remember that $ln(2) \approx 0.7$ and $ln(10) \approx 2.3$. (These are common bases to work in, so they're generally useful numbers.)
$$\begin{align} 5^n &> 4\ 000\ 000\\ \ln(5^n) &> \ln(4\ 000\ 000)\\ n (\ln 10 - \ln 2) &> 2 \ln 2 + 6 \ln 10\\ n (2.3 - 0.7) &> 2 \times 0.7 + 6 \times 2.3\\ 1.6 n &> 1.4 + 13.8\\ 1.6 n &> 15.2\\ 1.6 n &> 16 - 0.8\\ n &> 10 - 0.5\\ n &> 9.5\\ n &= 10 \end{align}$$
A bit much for mental arithmetic, but quite doable just typing into this here box.
The easiest way to multiply by $5$ without a calculator is to multiply by $10$ and then divide by $2$, i.e.: $$1: 5\times 5 = 50/2 = 25$$ $$2: 250/2 = 125$$ $$3: 1250/2 = 625$$ $$4: 6250/2 = 3125\ldots$$ Won't take you very long to get to $10$.
True. And for large power, use approximations :
$$5^{n} = \frac{10^n}{2^n} $$
$$5^9 = \frac{10^9}{2^9} = \frac{1,000,000,000}{512} \cong \frac{1,000,000,000}{500} \cong 2*10^6 < 4*10^6 $$
$$5^{10} = \frac{10^{10}}{2^{10}} = \frac{10,000,000,000}{1024} \cong \frac{10,000,000,000}{1000} \cong 10^7 > 4*10^6$$
It's not a mathematical way to prove, but it's a way to find the result using approximation.
The easiest way to multiply by 5 without a calculator is to multiply by 10 and then divide by 2. ie: 1: 5x5 = 50/2 = 25. 2: 250/2 = 125. 3: 1250/2 = 625. 4: 6250/2 = 3125... Won't take you very long to get to 10.
$$\log_{10}(5^n)=n\cdot \log_{10}(5)\approx n\cdot 0.7$$
$$\log_{10}(4000000)=\log_{10}(4)+6\approx 6.6$$
$$7\cdot 9=63\ \text{so that }\ \boxed{n=10}\ $$
$\log_{10}(2)\approx 0.3$ was only used giving $\log_{10}(5)=\log_{10}(10)-\log_{10}(2)$ and $\log_{10}(4)=2\cdot\log_{10}(2)$
(if non integer values are allowed $n\approx \frac {6.60206}{0.69897}$)
Taking square roots of both sides we solve $5^r>2000=25\cdot80$. The right side is approximated from below by $5^2\cdot5^2\cdot3$ so we want $5^{r-4}>3$ or $r=5$, so $n=2r=10$. Check $n=9$ is too small: $5^9<2^9\cdot 3^9<2^9\cdot 3^5\cdot 3^4=512\cdot 243\cdot 81<125,000\cdot 100<4,000,000$.
$5^n > 4,000,000$ find integer $n$.