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

How do I solve for $x$?

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

• is it $(.5)^{x-4}$ or $(.5)^x-4$? – Eleven-Eleven Oct 11 '13 at 18:27
• First write $(0.5)^{x-4}=2^{-(x-4)}$. – David Mitra Oct 11 '13 at 18:29

## 2 Answers

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

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

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

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