Contradicting Logs

Question

If $\log(10^2)=2$

and $(\log10)^2=(1)^2=1$

Then

$10^{\log(10^2)}=10^2=100$

But what about

$10^{(\log10)^2}$ ?

We get 2 solutions:

A. $10^{(\log10)^2}=10^{\log10} \times 10^{\log10}=10^1 \times 10^1=100$

B. $10^{(\log10)^2}=10^{1^2}=10^1=10$

What gives ?

Answer 1

You have missed the fact that $$a^x a^y = a^{x+y},$$ not $$a^x a^y \ne a^{xy}.$$ Hence, in (A), $$10^{(\log 10)^2} \ne 10^{\log 10} 10^{\log 10} = 10^{2 \log 10}.$$

Answer 2

Hint:

$(a^x)^y \ne a^{x^y}$

(you have proved this fact)

Answer 3

Your question is "What gives?"

$$x^{(y^z)}\ne {(x^y)}^z$$

For example,

$$2417851639229258349412352=2^{(3^4)}\ne {(2^3)}^4=4096$$ This is essentially the fallacy in lines A. and B.. The logarithms obscure this.