Finding the nth Exponent: Solving for n in 2^2^n = 36 trillion

[MrAnderson]

Homework Statement

find n.

Homework Equations

2^2^n = 36trillion

The Attempt at a Solution

I have worked this out and found n to be equal to 3.67 can someone confirm that i have the right solution? I took the log on both sides and worked it out.

 

[Mark44]

I get n ~ 5.4929. You have to take the log of both sides twice.
For my value of n, 2^n ~ 45.03, and 2 to that power is close to 36 trillion.

 

[MrAnderson]

I get n ~ 5.4929. You have to take the log of both sides twice.
For my value of n, 2^n ~ 45.03, and 2 to that power is close to 36 trillion.

That's correct that 2^n = 36trillion and n=45, i get this also. But when i attempt to get n when 2^2^n = 36 trillion i first convert 2^nlog2 =log(36trillon)
which becomes log 2^nlog2 =log(log(36 trillion)), now at this point i divide both sides by log2. Now I am totally stuck and lost. Can you explain the process?

 

[Mark44]

2^(2^n) = 36 x 10^12
==> (2^n) log 2 = log(36 x 10^12)
==> n log(2) + log(log 2) = log(log(36 x 10^12))

Now, subtract log(log 2) from both sides, and then divide both sides by log(2).

I think you did something weird with the exponents when you got 2^nlog2. The exponent on 2 of your original express is 2^n, so when you take the log of 2^2^n, you get 2^n * log 2, and not what you show.

 

[MrAnderson]

2^(2^n) = 36 x 10^12
==> (2^n) log 2 = log(36 x 10^12)
==> n log(2) + log(log 2) = log(log(36 x 10^12))

Now, subtract log(log 2) from both sides, and then divide both sides by log(2).

I think you did something weird with the exponents when you got 2^nlog2. The exponent on 2 of your original express is 2^n, so when you take the log of 2^2^n, you get 2^n * log 2, and not what you show.

Oh i see, you have to add them and then subtract them then divide by log(2). thanks a lot, you've been very helpfull :).