Exponents and what they really mean

Question

I have two questions:

1. I need to solve for x here

$$0.95 = \exp(-(1+0.4\frac{x-20}{4})^{-\frac{1}{0.4}})$$

My steps:

$$\ln(0.95) = -(1+0.4\frac{x-20}{4})^{-\frac{1}{0.4}}$$

$$\ln(0.95)^{-0.4} = -(1+0.4\frac{x-20}{4})$$

Then I punch in calculator $\ln(0.95)^{-0.4} = 3.2808$. Negating and subtracting $1$, multiplying by $4$ dividing by $0.4$ and add $20$, I get $-22.8077$. However, the right answer is $42.8077$.

I thought I performed all steps correctly, so where did I go wrong?

2. A more general question regarding how exponents work. I know that:

$$a^{b} = \exp(\ln(a)*b)$$

And it makes sense that, $$(-5)^{0.5} = \exp(\ln(-5)*0.5)$$ is undefined.

However, when I punch in, $$(-5)^{0.4} $$ to my TI-30XS Multiview calculator, I got 1.903653939. No $i$ what so ever.

So I guess I don't know how exponents work any more.

Answer 1

From here

$$\ln(0.95) = -\left(1+0.4\frac{x-20}{4}\right)^{-\frac{1}{0.4}}$$

we have

$$-\ln(0.95) = \left(1+0.4\frac{x-20}{4}\right)^{-\frac{1}{0.4}}$$

and then

$$[-\ln(0.95)]^{-0.4} = \left(1+0.4\frac{x-20}{4}\right)$$

Answer 2

The negation needs to come before the exponentiation. This will turn your $-22.8076983$ into a positive $22.8076983$. Plus, you don't need to deal with complex numbers.

Have you ever seen the equation $e^{\pi i} = -1$? Have you ever noticed that exponentiation always results in a positive number when you're working with the "Real" number line? What happened was you were trying to invert an exponentiation with a logarithm, but since it's negative, that only makes sense in the "Complex" number system. Complex logarithms can end up having multiple answers ("branches"?), though I'm not sure if that's what you encountered there.

Answer 3

Let's first simplify by setting $$ y=1+0.4\frac{x-20}{4}=1+\frac{x-20}{10}=\frac{x}{10}-1 $$ so that your expression becomes $$ 0.95=\exp(-y^{-1/0.4}) $$ Taking logarithms: $$ \ln0.95=-y^{-1/0.4} $$ Negate and raise both sides to $-0.4$: $$ (-\ln0.95)^{-0.4}=y $$ With a calculator, $$ y=3.280769831611707 $$ Therefore $$ x=10(y+1)=42.807698316117066 $$

Note that $\ln0.95<0$, so raising it to $-0.4$ is not really the best thing to do: better dealing with positive numbers.