how large do i have to make?

Question

How large do you need to make x so that

e ^ (-x / 5) < 0.1

Answer 1

e^(-x/5) < 0.1

The function f(x) = ln(x) is an increasing function, which means if
a < b, then f(a) < f(b). Therefore,

f(e^(-x/5)) < f(0.1)

ln (e^(-x/5)) < ln(0.1)

By the log property,

(-x/5) ln(e) < ln(0.1)

ln(e) = 1, so we get

(-x/5) < ln(0.1)

Multiplying by (-5) flips the inequality but isolates the x, giving us

x > (-5) ln(0.1)

-5 ln(0.1) approximates to 11.513

So a value greater than 11.513 will suffice.

Answer 2

e ^(-x/5) < 0.1
LN(e^(-x/5)) < LN(0.1)
-x/5 < -2.30259
-x < -11.51293
x > 11.51293
Approximately, I've obviously rounded a little.
The exact answer would be:
x > -5*LN(0.1)
-or-
x > LN(100,000)

Answer 3

Infinitely large. Since negative infinity/5 is negative infinity, and since you're raising e to negative infinity, then it would approach zero. It's a simple limit concept.

Answer 4

Take logs on both sides and u'll find the value for x

Answer 5

what?