I have a variable as an exponent, how do I isolate it from everything else?

[kelp]

Homework Statement

.75 = 1 - (1 - p)^n
I want to get the n alone, and find n in terms of p.

Homework Equations

The Attempt at a Solution

I simplified it to this:
1 - p = (.25)^(1/n)
Basically, I just moved numbers around. I do not know how to get the n by itself without dragging along a number.

Thanks!

 

[Mark44]

Homework Statement

.75 = 1 - (1 - p)^n
I want to get the n alone, and find n in terms of p.

Homework Equations

The Attempt at a Solution

I simplified it to this:
1 - p = (.25)^(1/n)
Basically, I just moved numbers around. I do not know how to get the n by itself without dragging along a number.

Thanks!

Take the log of both sides.

.75 = 1 - (1 - p)ⁿ
==> .25 = (1 - p)ⁿ
==> ln(.25) = ln[(1 - p)ⁿ]

Using one of the properties of logarithms, you can work with the right side to eventually isolate n.

 

[Char. Limit]

Have you tried logarithms?

 

[zketrouble]

Logarithms would be the best way to get change the variable from a constant to a coefficient. Any logarithm will work, whether it be the standard log₁₀, the natural log ln, or any other one.

Just keep in the back of your head that:

log(a^x) = x*log(a)

Here's an example on how to use this property:

3.2^x = 10.24
ln(3.2^x) = ln(10.24)
Using the property mentioned:
x*ln(3.2) = ln(10.24)
x*1.1631 = 2.3263
x = 2.3263/1.1631
x = 2

Note (again) that it doesn't matter whether you choose the ln or log₁₀ or log₁₂₃₄₅, so long as you use the logarithm on BOTH sides of the equation and then use the mentioned property you'll be fine.