Question

(Algebra) I just realized that when exponentiating some value and using properties of exponents, some things come up that make no sense to me; what's the "illegal" algebra move here? (Math below)

Answer 1

If I have a statement like \[\frac{\alpha '}{\alpha} =\frac{2}{x}\]

Answer 2

If I use separation of variables, take the natural log of both sides and eventually end up with: \[\ln(\alpha)=2\ln(x)\]

Upon exponentiating, I can get:\[\alpha = e^{2 \ln(x)}\]

Answer 3

Why, when I separate the two multiplied terms in the exponent, e.g. \[\alpha = e^2 e^{\ln(x)}=e^2 x = e^{2 \ln(x)}=e^{\ln(x^2)}=x^2\]

Answer 4

Why are these statements not equivalent? Because e^2x =/=x^2, obviously.

Answer 5

\[a^{mn}\ne a^m\cdot a^n\]

Answer 6

\[a^{mn} = \left(a^m\right)^n\]

Answer 7

Oh god, what am I thinking of? In what context is something like that similar? There's something like that, but I can't recall what. Not just e^m^n, either

Answer 8

Huh, well thanks, anyways! I don't know what I'm remembering, but there's some other similar property or situation I'm failing to recall.

Answer 9

when dealing with integers, we can think of exponentiation as a short cut for repeated multiplication
\[a^m = a\cdot a\cdot a\cdot \cdots (\text{m times})\]

Answer 10

\[\color{blue}{a^n = a\cdot a\cdot a\cdot \cdots (\text{n times})}\]

Answer 11

multiplying them both gives
\[a^m\bullet\color{blue}{a^n}~~=~~a^n = a\cdot a\cdot a\cdot \cdots (\text{m times})\bullet \color{blue}{a^n = a\cdot a\cdot a\cdot \cdots (\text{n times})}\]

it is easy to see that \(a\) is getting multiplied \(m+\color{blue}{n}\) times on right hand side

Answer 12

therefore
\[a^m\bullet\color{blue}{a^n}~~~=~~~a^{m+\color{blue}{n}}\]

Answer 13

we can try and make sense of the other rule \((a^m)^n =a^{mn}\) also with similar reasoning

Answer 14

**
\[a^m\bullet\color{blue}{a^n}~~ =~~ a\cdot a\cdot a\cdot \cdots (\text{m times})\bullet \color{blue}{a\cdot a\cdot a\cdot \cdots (\text{n times})}\]