Question on the ambiguity of repeated exponentiation

Question

I know that when there is an exponent to an exponent, you just multiply the exponents, for instance $a^{b^c}=a^{bc}$.

However, in this case:

$$\log_a b^{c^d} =\log_a b^{cd}$$

I can say this due to the property I initially mentioned.

If the above statement is correct, this would mean

$$c^d*\log_a b =cd*\log_a b$$

I know this is not always true, since $c^d=cd$ would always be true if this was the case.

I can disprove this with a counterexample, but I do not know where I went wrong.

Can someone please explain my mistake in a mathematical way, and what I should have done, instead (of my incorrect step)? Thank you.

This is my first time on this site, I tried my best on the formatting.

Answer 1

To add on to the other answers: because $(a^b)^c$ simplifies to $a^{bc}$ while $a^{(b^c)}$ does not simplify to anything, the convention is to take $a^{b^c}$ to mean $a^{(b^c)}$: the version that we can't write just using multiplication. (You could just memorize the rule, but it helps to understand why the rule was adopted.) In general, if you have a tower of exponents $$a^{b^{c^{d^{e^f}}}}$$ you start evaluating it from the top down: first find $e^f$, then $d^{(e^f)}$, then $c^{(d^{(e^f)})}$, and so on.

This version also results in much bigger numbers, which is always fun. For example, $(3^3)^3$ is just $3^9 = 19\,683$. On the other hand,$3^{3^3}$ is $3^{27} = 7\,625\,597\,484\,987$.

Answer 2

$$(a^b)^c = a^{bc}$$ but in general $$a^{b^c} \neq a^{bc}$$

Answer 3

Note that in general $$a^{b^c}\neq a^{bc}$$

for the equality we should write

$$(a^b){^c}=a^{bc}$$

and

$$\log_a (b^c){^d} =\log_a b^{cd}=cd\log_a b$$

Answer 4

$a↑(bc) = a↑b↑c \neq a^{(b^c)} = a^{b^c}$ except for special values for a, b and c.

For instance:

$64 = 2↑(2 \times 3) = 2↑2↑3 \neq 2^{(2^3)} = 256$

Exponential towers are used to create very large numbers.

$b^{cd}$ is not equal to $b^{c^d}$ for the same reason.