Fundamental Identities on Exponents

Question

I came across some properties of "Exponential and Logarithmic Equations and Inequalities" in the book "Problems in Mathematics" by V GOVOROV. The problem which I am facing is of one of the properties in it.

The property given in book goes like this:

$a^x \cdot b^x = (a\cdot b)^x, b>0$

I understand the meaning of the first part of property but what is the need of condition "b>0" in the property. The property still holds true if assumed "$a=2; b=-3; x=4$"

$$(2)^4 \cdot (-3)^4 = (16) \cdot (81) = 1296$$

$$(2)^4 \cdot (-3)^4 = ({2\cdot-3})^4 = (-6)^4 = 1296$$

So why is the condition "$b>0$" required?

Answer 1

Because in general $b^x$ only makes sense if $b\geq0$. There are a few exponents that work for negative numbers, namely integer powers. But in general one defines $b^x=e^{x\log b}$, which requires $b>0$. There is no natural meaning for something like $(-2)^{0.7}$.

Answer 2

the condition is necessary when x is not an integer but a fraction or decimal
for eg. Take $a=1, b=-1, x=0.1234$
now the result would be $$1^{0.1234}.(-1)^{0.1234}=(-1)^{0.1234}$$
which is undefined in real numbers hence the condition holds.
I think the condition may be $ab \gt 0$