How do I solve this exponential function? $2^{-100x} = (0.5)^{x-4}$

Question

How do I solve for $x$?

$2^{-100x} = (0.5)^{x-4}$

Answer 1

Hint: you need the same base: So $0.5=\frac{1}{2}=2^{-1}$

Thus, $2^{100x}=(2^{-1})^{x-4}=2^{4-x}$

Answer 2

$$ -100x\ln\left(2\right) = \left(x - 4\right)\ln\left(0.5\right) = x\ln\left(0.5\right)- 4\ln\left(0.5\right)\,, \quad x = {4\ln\left(0.5\right) \over 100\ln\left(2\right) + \ln\left(0.5\right)} $$

$\ln\left(0.5\right) = -\ln\left(2\right)$ $$ \color{#ff0000}{\large x} = {-4\ln\left(2\right) \over 100\ln\left(2\right) - \ln\left(2\right)} = \color{#ff0000}{-\,{4 \over 99}} $$