Is my algebra correct here?

Question

so I have the interest equation

$$P=C(1+\frac rn)^{nt}$$

and I need to solve for t. So what I did was divide to get rid of C first:

$$\frac PC=(1+\frac rn)^{nt}$$

then I took the logarithm to get nt on it's own

$$\log_{1+\frac rn}\frac PC = nt$$

divide by n...

$$\frac {\log_{1+\frac rn}\frac PC}n = t$$

Then the change of base to natural logs

$$t=\frac {\ln (\frac PC)}{n\ln (1+\frac rn)}$$

I was curious because Wolfram Alpha gave me a different answer so I wasn't sure if there's something I'm doing wrong that I should fix in the way I do algebra. I'm taking calculus next year so it's important that I do things correctly.

I would take the natural logarithm, or your favourite logarithm, immediately. So no passing through the logarithm to a weird base. But the result would be the same, just a shorter calculation. Note that $\ln((1+r/n)^{nt})=nt \ln(1+r/n)$. — André Nicolas, Feb 22 '16 at 05:57
@AndréNicolas Ah yes, I forgot about the exponentiation rule. Thank you — AJB_1070179, Feb 22 '16 at 06:14

score 1 · Accepted Answer · answered Feb 22 '16 at 06:26

1

Your algebra is fine, but use the natural logarithm first, so:

$$\frac PC=(1+\frac rn)^{nt}$$ $$\ln\frac PC=\ln[(1+\frac rn)^{nt}]$$ $$\ln\frac PC=nt\ln(1+\frac rn)$$ $$t=\frac{\ln(\frac PC)}{n\ln(1+\dfrac rn)}$$

Wolfram|Alpha sees the $C$ as the Fresnel Integral (look at the bottom right of the top box for 'solve P=C(1+r/n)^(nt) for t'.

answered Feb 22 '16 at 06:26

JMP

21,771

that's okay$$$$ – JMP Feb 22 '16 at 06:29

Is my algebra correct here?

1 Answers1