Logarithms: How to find a power of a power?

[bardothello]

Hello,

I was thinking about logarithms again, and realized that I never learned how to manipulate an exponent of an exponent.

If I have:

2^(2^x)=256, (I don't know how to give an exponent to a power in teX format),

How do I go about finding what x is? Can I use logarithms?

I tried finding an answer online but came up short. Also, is there a special term for referring to a power of a power?

Thanks in advance!

 

[MarkFL]

I would write:

[MATH]2^{2^x}=2^{2^3}[/MATH]
This implies:

[MATH]x=3[/MATH]

 

[topsquark]

@bardothello:
MarkFL has it right for your specific problem, which you did fine with when you wrote it. However you need to be aware that the notation for the power of a power is not standard. We have two possible meanings for something like [math]x^{y^z}[/math]. This could be interpreted as either [math]x^{(y^z)}[/math] or [math]\left ( x^y \right ) ^z[/math].

-Dan

 

[bardothello]

alright, thanks guys!

 

[MarkFL]

alright, thanks guys!

Consider the more general problem:

Solve for \(x\):

[MATH]a^{b^x}=c[/MATH]
This implies:

[MATH]b^x=\log_a(c)[/MATH]
And this in turn implies:

[MATH]x=\log_b\left(\log_a(c)\right)[/MATH]
Now, if we then apply this to the problem you posted, we have:

[MATH]a=b=2,\,c=256=2^8[/MATH]
And we obtain:

[MATH]x=\log_2\left(\log_2\left(2^8\right)\right)=\log_2\left(8\right)=\log_2\left(2^3\right)=3[/MATH]

 

[lookagain]

@bardothello:
However you need to be aware that the notation for the power of a power is not standard. We have two possible meanings for something like [math]x^{y^z}[/math]. This could be interpreted as either [math]x^{(y^z)}[/math] or [math]\left ( x^y \right ) ^z[/math].

-Dan

No, it is standard, and it has one meaning. It is top-down exponentiation.

Notice in MarkFL's post #5 general case. He went right to the top-down form
when first applying logarithms to each side.

Another example: \(\displaystyle \ \ 3^{3^3}\) does not have two meanings. It is equal to
\(\displaystyle 3^{27}\). It is not equal to \(\displaystyle 27^3\).

This is in contrast to a^b^c, which should never be used, because it is
ambiguous/meaningless. There, in that horizontal style, either a^(b^c) or
(a^b)^c should be written, respectively, to indicate the different exponentiation.

 

[Dr.Peterson]

See Wikipedia: https://en.wikipedia.org/wiki/Exponentiation#Identities_and_properties:

Without parentheses, the conventional order of operations in superscript notation is top-down (or right-associative), not bottom-up[13] (or left-associative). That is, [MATH]b^{p^{q}}=b^{\left(p^{q}\right)}[/MATH], which, in general, is different from [MATH]\left(b^{p}\right)^{q}=b^{pq}[/MATH].​

Also, https://en.wikipedia.org/wiki/Order_of_operations#Serial_exponentiation:

If exponentiation is indicated by stacked symbols using superscript notation, the usual rule is to work from the top down:[1][11]​

​

a^bc = a^(bc)​

​

which typically is not equal to (a^b)^c.​

​

However, when using operator notation with a caret (^) or arrow (↑), there is no common standard. For example, Microsoft Excel and computation programming language MATLAB evaluate a^b^c as (a^b)^c, but Google Search and Wolfram Alpha as a^(bc). Thus 4^3^2 is evaluated to 4,096 in the first case and to 262,144 in the second case.​

The standard typeset appearance of such a tower of exponents, with the exponents decreasing in size, illustrates this interpretation!

On the other hand, it is not commonly taught explicitly as part of the "order of operations" in initial teaching, so it is not as well-known as other such facts.