Thread: How many tens?

Can someone please help with this question please? Thanks!

3^667 * 2^2 = X * 10^Y

I need to find X and Y. I.e. convert to scientific notation. Thanks!

And this one too.

2^10^6 = X * 10^Y

Find X & Y.

I don't know if my answer is good enough for you.

As I learned from limit taking techniques of calculus, when there are power look like things in the equation, take log on both sides won't change the equality of the equation.

For the second equation of your question, take natural log on both sides, get:

ln(2^10^6)=ln(x*10^y)
ln(10^6)/ln(2)=ln(x)+ln(10^y)
ln(10^6)/ln(2)=ln(x)+y*ln(10)

now you have a y times something

Nope that wasn't what I was looking for. Thanks anyway.

3^667 * 2^2 = X * 10^Y
I need to find X and Y. I.e. convert to scientific notation. Thanks!

And this one too.
2^10^6 = X * 10^Y
Find X & Y.

Let us derive some formulas.

10^a = b ----------------------------------(i)
Take the common logs of both sides,
a*log(10) = log(b)
a*1 = log(b)
a = log(b)
Substitute thjat back into (i),
10^(log(b)) = b
Or,
b = 10^log(b) -----------(ii), where log is logarithm to the base 10.

So,
3^667 = [10^log(3)]^667 = 10^[667*log(3)]
2^2 = [10^log(2)]^2 = 10^[2*log(2)]
Then,
(3^667)*(2^2)
= 10^[667log(3)] *10^[2log(2)]
= 10^{667log(3) +2log(2)}
Using a calculator,
= 10^318.8419369 -----***
= 10^318 *10^0.8419369
= 10^318 *6.949233427
= 6.949233427 *10^318 ----------***

Therefore, in [3^667 * 2^2 = X * 10^Y],
X = 6.949233427, and Y = 318. ---------------answer.

---------------
2^10^6 = X * 10^Y

2^10^6
= (2^10)^6
= 2^(10*6)
= 2^60 -------------***
= [10^log(2)]^60
= 10^ 60log(2)
= 10^ 18.06179974
= 10^18 *10^0.06179974
= 10^18 *1.152921505
= 1.152921505 *10^18 --------***

Therefore, in [2^10^6 = X * 10^Y],
X = 1.152921505, and Y = 18. -----------------answer.

Thanks a lot ticbol!