Why is $\left(\frac{1}{2}\right)^{x} = \frac{1}{7}$ the same as saying: $(2)^{x} = 7$

Question

Why is $\left(\frac{1}{2}\right)^{x} = \frac{1}{7}$
the same as saying: $(2)^{x} = 7$

Sorry for the really dumb question but I'd like to see the process of how this is achieved.

Answer 1

Let's say we have two fractions in an equality: $$\frac{a}{b}=\frac{c}{d}$$ where $b\neq0\neq d$. We can flip them over, and say that $$\frac{b}{a}=\frac{d}{c}$$ where $a\neq0\neq c$.

Let's apply that here. First, we distribute out the $x$: $$\left(\frac{1}{2}\right)^x=\frac{1^x}{2^x}$$ Now we substitute this back in: $$\frac{1^x}{2^x}=\frac{1}{7}$$ Flipping this, we have $$\frac{2^x}{1^x}=\frac{7}{1}$$ To finish, think about what $1^x$ equals, and what $\frac{7}{1}$ simplifies to.

Answer 2

$7 = \frac{1}{1/7} = \frac{1}{(1/2)^x} = \left(\frac{1}{2}\right)^{-x} = \frac{1}{2^{-x}} = 2^x$.

Answer 3

$$ \left(\frac{1}{2}\right)^x = \frac{1}{2^x} $$

Then if you just cross multiply you get $2^x = 7$.

Answer 4

Because $(\frac a b)^x = \frac {a^x} {b^x}$.

With that in mind $(\frac 1 2)^x = \frac {1^x}{2^x} = \frac 1 {2^x} = \frac 1 7$

and therefore $\frac 1 {2^x}*[2^x*7] = \frac 1 7*[2^x*7]$ and therefore $7 = 2^x$.