# Thread: How to solve for an exponent...

1. ## How to solve for an exponent...

Can you show me how to solve for n in the equation:

60000= 38400(1.07)^n

Thank you.

2. Originally Posted by oryxncrake
Can you show me how to solve for n in the equation:

60000= 38400(1.07)^n

Thank you.
First simplify a bit

$\displaystyle 25=16(1.07)^n$

Now take nat. logs of both sides

$\displaystyle \ln{25}=\ln{\left[16(1.07)^n\right]}$

Can you proceed?

3. Originally Posted by oryxncrake
Can you show me how to solve for n in the equation:

60000= 38400(1.07)^n

Thank you.
Isolate the base and the exponent using division and then take the logarithm using the property $\displaystyle ln(a^k) = k\, ln(a)$

Spoiler:

$\displaystyle 1.07^n = \frac{60000}{38400} = \frac{25}{16} = (\frac{5}{4})^2$

That last part will make sense next step

$\displaystyle n\, ln(1.07) = 2\, ln(1.25)$

$\displaystyle n = \frac{2ln(1.25)}{ln(1.07)} = 6.60 (3sf)$

4. Another way...

$\displaystyle 60000=38400(1.07^n)$

$\displaystyle \frac{60000}{38400}=1.07^n$

$\displaystyle 1.07^n=\frac{25}{16}$

$\displaystyle \ln 1.07^n=\ln \frac{25}{16}$

$\displaystyle n \ln 1.07=\ln 25- \ln 16$

$\displaystyle n=\frac{\ln25-\ln16}{\ln1.07}$