How do you find the inverse of #y=(1/2)^x#?

Question

Christian Antunez · Accepted Answer

undefined To find the inverse of $ y = \left(\frac{1}{2}\right)^x $, follow these steps:

1. Replace $ y $ with $ x $ and $ x $ with $ y $ to interchange the dependent and independent variables: $ x = \left(\frac{1}{2}\right)^y $.
2. Solve the new equation for $ y $.
3. To solve for $ y $, take the logarithm of both sides of the equation. Since the base is $ \frac{1}{2} $, you should use the logarithm base $ \frac{1}{2} $ or, equivalently, the natural logarithm (ln) which will give the same result.
4. After taking the logarithm, isolate $ y $ to find the inverse function.

So, starting with $ x = \left(\frac{1}{2}\right)^y $:

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

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

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

$$ \ln(x) = -y \ln(2) $$

$$ y = -\frac{\ln(x)}{\ln(2)} $$

Therefore, the inverse of $ y = \left(\frac{1}{2}\right)^x $ is $ y = -\frac{\ln(x)}{\ln(2)} $.

Elias Ackerman · Answer

#color(white)xxx=-log_2 y# #color(white)xxy=(1/2)^x# #=>x=log_(1/2) y# #=>x=log_(2^-1) y# #=>x=-log_2 y#