4000(1.5^x) = 25,000?

Yup, it's a log.

So how exponents work is this:

base^exponent=solution

Using those same terms, a log would be written like this:

logbase(solution)=exponent.

For example, 5^3=125

that can be rewritten as log5(125)=3.

So in your case, you would say log1.5(6.25)=x.

Now if you don't have a calculator which can use a base other than 10 (with the log button), then you would use the change of base formula which is this:

logb(x) = (log(x))/log(b)).

In your case, you would write (log(6.25))/(log(1.5)), to get an answer of approx 4.519702009.

Hope I helped :)

Step one: Divide both sides by 4000.

Step 2: Take the ln of both sides.

xln1.5=ln6.25.

Divide both sides by ln 1.5.

x=4.5197

log base 1.5 of 6.5 = 4.5197020091

I don't really understand how logorithms work either, or fractional exponents.

1.5*1.5*1.5*1.5(1.5*0.5197020091)

Does not equal

1.5^4.5197020091

Which is weird to me.

Divide both sides by 4000.

1.5^x = 6.25

x(log(1.5)) = log(6.25)

x = (log(6.25))/log(1.5)

x ≈ 4.5197

If you're confused about the math, ask and I'll give you an explanation.

1.5^x=25000/4000=6.25

xln1.5=ln6.25

x=ln6.25/ln1.5=4.519702009

All are correct !!