Which is larger?

$\LARGE \color{#D61F06}{10}^{\color{#3D99F6}{7}} \quad \text{or} \quad \color{#3D99F6}{7}^{\color{#D61F06}{10}}$

$10^7$
$7^{10}$
Equal

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$a=10^{7},b=7^{10} , \frac{a}{10^{7}}=1, \frac{b}{10^{7}}= \frac{7^{10}}{10^{7}}= \frac{7^{3}*7^{7}}{(10)^{7}}= 7^{3} * (\frac{7}{ 10})^{7}$ >1 then b>a

Muh Ali
- 5 years, 7 months ago

Otto brestcher......how did you observe this amazing fact

Bishwayan Ghosh
- 2 years, 5 months ago

Log in to reply

More than 15k observed this fact from last 12000 years of dressed tail less ape civilisation

Theerdala Vignesh
- 6 months, 1 week ago

Wow that's a brilliant answer

Sareeta Devi
- 2 years, 3 months ago

take log , log 10^7 = 7 log 10 = 7 x 1 = 7 and log7^10 = 10 log 7 = 10 x 0.84 = 8.45

so B > A

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kindly dear Roy , let me know how you reached the log values?

Jafir Khan Niazi
- 5 years, 9 months ago

please tell me why you multiply 0.84 with 10??

Ariful Ashiq
- 5 years, 9 months ago

I did a similar method but didnt find the values of the log

Diyah Muhammed
- 4 years, 6 months ago

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$2^7 \times 2^{18} \color{#D61F06}\geq 2^7 \times 32^{3.6}$

Cleres Cupertino
- 5 years, 10 months ago

Excellent!

Exponent Bot
- 3 years, 2 months ago

7^10=49^5*

Adit Goud
- 5 years, 10 months ago

$7^{10} = 282475249, 10^{7} = 10000000$

$7^{10} > 10^{7}$

Actually, this is easier!

汶良 林
- 5 years, 10 months ago

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Sure, if you have a calculator and trust it and yourself not to make mistakes.

Whitney Clark
- 5 years, 10 months ago

Consider the function, y=x^{1/x}. Evaluate dy/dx and it turns out to be x^{(1/x)-2} (1-ln x), which when equated to zero gives x=e. Note that dy/dx >0 for x <e, dy/dx=0 for x=e and dy/dx <0 for x>e. Therefore the function y=x^{1/x} is a decreasing function when x>e. Hence, for x1>x2>e, x2^{1/x2} > x1^{1/x1}.

Since 10>7>e, 7^(1/7) > 10^(1/10). Raisiing the power to 70 on both sides, 7^10 > 10^7

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7x7x7x7x7x7x7x7x7x7=282475249>10x10x10x10x10x10x10=1000000

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Simply 10^7 is (0.01)^10 which is less than 7^10 :) :)

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This is funniest logic here. XD Go back to 6th grade

Saurabh Sharma
- 5 years, 8 months ago

10^7 is not the same as (0.01)^10...

Jana Carpinato
- 4 years, 4 months ago

282475249>10000000 probably B is correct

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For $10^7$ there will be seven zeroes after one

$10^7 = 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 10000000$

Now,

$7^{10} = 49^5$ or $282475249$

Therefore $7^{10}$ is larger.

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i did it in the same way

Halima Tahmina
- 4 years ago

thanks.your comment inspired me

Mohammad Khaza
- 4 years ago

If you know that $7^3 = 343$ , then you can estimate that $7^6 = 343^2$ is a little greater than $100000 = 10^5$ .

With that in mind, base 10 logarithms tell us that $\log(7^6)\ge\log(10^5)\;\Rightarrow\;6\log7\ge5\;\Rightarrow\;\log7\ge\frac56$ .

Thus, $\log(7^{10}) = 10\log7 \ge 10\cdot\frac56 = \frac{25}3$ .

Since $\log(10^7) = 7 < \frac{25}3$ , then $\log(10^7) < \log(7^{10})$ , which means $7^{10} > 10^7$ .

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the best answer i saw ..u don't have to use a calculator but not a quickly one :D

Ibrahim Said
- 5 years, 8 months ago

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= $7^{10} ... 10^{7}$ , = $7^{3}$ ... $10$ , $= 343 > 10$ . So, $7^{10} > 10^{7}$

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Since power is bigger number of significant figures is more thus 7^10 is bigger than 10^7.

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7^10>6^10=6^7×6^3 6^3=2^3×3^3>2^3×2^4=>2^7 Therefore, 7^10>6^10>6^7×2^7=12^7>10^7 Hence proved

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I cheated and used the fact that I have the first ten powers of 2 memorized:

4<7<8, so (4^10)<(7^10)<(8^10)

4^10=(2^10)^2 or roughly 10^6

8^10=(2^10)^3 or roughly 10^9

7 is closer to 8, so 7^10 is going to be closer to 10^9 which is larger than 10^7

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10 exponent 7 = 100M 7 exponent 10 =70B therefore 7 exponent 10 is larger than 10 exponent 7.

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$x_1, x_2$ , If $x_1, x_2>e$

$\Rightarrow$ If $x_1>x_2$ then $x_2^{x_1}>x_1^{x_2}$ and vice versa...

Kishore S. Shenoy
- 5 years, 8 months ago

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Check this one...... Log 10^7 < log 7^10...... 7log 10 < 10 log 7 ...... 7x1 < 10x0.84

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10^7 ? 7^10

1 ? .7^7*7^3

1? (.7)*(3.43)^3

70% of 3 itself is greater than 1

So 1< (.7)*(3.43)^3

Hence 7^10 is bigger.

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We know that 7^8 > 10^7, and therefore 7^10 >> 10^7. Thats it

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Let's put "?" for our relation.

10⁷ ? 7¹⁰ log(10⁷) ? log(7¹⁰) [apply log, which is increasing, so the "?" sign stays the same] 7.log(10) ? 10.log(7) [log property] 7 ? 10.log(7) [log(10=1)]

Now, log(7)>1, so, 10 log(7)>10, therefore 7 < 10.log(7) which means that "?" = "<" and then

10⁷ < 7¹⁰

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10^7=(7x1.43)^7 so divide both by 7^7 so 1.43^10 versus7^3 is no brainer 7^3 is much larger

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$7^{10}=49^{5}>40^5=4^5\times10^5>10^7$ since $4^5=1024$