Question

Trying to solve for 'n'.
(r*pv)/pmt = 1-((1+r)^-n)
if r=.01; pv=4,495.50;pmt=100;n=60 then
(.01*4,495.50)/100 =.44955
and
1-((1+r)^-n) equals
1 -(1.01)^-60 equals 0.44955
I am trying to solve the equation for 'n'
which requires taking logs of both sides.
The log of (r*pv)/pmt is easy enough to find but trying to find the log of the right side of the equation 1-((1+r)^-n) is a lot more difficult.
Ordinarily the log should be -n* (1 - (1+r)).
However, the "1" the "-(1+r)" and the "-n" are presenting problems. (Negative numbers present severe problems when dealing with logarithms.
Anyone have any idea how to solve the equation for "n"?

Answer 1

\(\dfrac{r \times pv}{pmt} = 1-(1+r)^{-n}\)

Add the quantity with n in the exponent to both sides and subtract the left fraction from both sides to have all terms with n on the left side:

\((1+r)^{-n}= 1-\dfrac{r \times pv}{pmt} \)

1 = pmt/pmt, and add the fractions onthe right side.

\((1+r)^{-n}= \dfrac{pmt - r \times pv}{pmt} \)

Take logs of both sides:

\(\log [(1+r)^{-n}]= \log[\dfrac{pmt - r \times pv}{pmt}] \)

\(-n\log (1+r)= \log[\dfrac{pmt - r \times pv}{pmt}] \)

\(-n= \dfrac{\log[\dfrac{pmt - r \times pv}{pmt}]}{ \log (1+r) } \)

\(-n= \dfrac{\log( pmt - r \times pv) - \log pmt}{ \log (1+r) } \)

\(n= \dfrac{\log pmt -\log( pmt - r \times pv) }{ \log (1+r) } \)

Using your values, and solving for n:

\(n= \dfrac{\log 100 -\log( 100 - 0.01 \times 4495.5) }{ \log (1+0.01) } \)

\(n= \dfrac{2 -\log 55.045 }{ \log (1.01) } \)

\(n = 59.99992989\)

n comes out very close to 60.

Answer 2

Wow thanks. Yes, that was a little bit "beyond" me.

Answer 3

yw