Compound Interest, Continuous Compounding & Logarithms

Compound-Interest Fundamentals

  • Present value (initial principal) denoted P.

  • Annual percentage yield (APR) denoted r (expressed as a decimal; e.g.
    15 % ⇒ r = 0.15).

  • Interest is reinvested (compounded) at the end of each compounding period.

  • 1-year accumulation with annual compounding:

    • A(1)=P+Pr = P(1+r)=P\bigl(1+0.15\bigr) when r=0.15.

  • 2-year accumulation (still annual compounding):

    • Second-year interest is earned on BOTH the original P and the interest from year 1.

    • A(2)=P(1+r)(1+r)=P(1+r)^2.

  • In general, after t years with annual compounding:

    • A(t)=P(1+r)^t — exponential growth where

    • coefficient A=P,

    • base B=1+r.

Compounding m Times per Year (Periodic Compounding)

  • Each period’s rate = \dfrac r m.
    – Ex: Quarterly compounding ⇒ m=4\;\Rightarrow\;\text{rate per quarter}=r/4.

  • Number of periods in t years = mt.

  • General periodic-compounding formula:

    • \displaystyle A(t)=P\left(1+\frac{r}{m}\right)^{mt}.

  • Special case annual compounding: m=1 gives A(t)=P(1+r)^t again.

Worked Example 1 – $2000 at 12.6 % Compounded Monthly

  • Given: P=2000\,,\;r=12.6\%=0.126\,,\;m=12\text{ (monthly)}\,,\;t=2.5\text{ yr}.

  • Plug into periodic formula:
    \displaystyle A(2.5)=2000\Bigl(1+\tfrac{0.126}{12}\Bigr)^{12\times2.5}=2000\,(1.0105)^{30}\approx 2793.!97.

  • Calculator entry tip:
    2000*(1+0.126/12)^(12*2.5).

Building an Explicit Exponential Model & Goal-Time Estimate

  • Model form: A(t)=ab^{t}.
    – Here a=P=2000.
    – b=(1+0.126/12)^{12}=1.0105^{12}\approx1.126.

  • Final model: \displaystyle A(t)=2000\,(1.0105)^{12t}.

  • Estimate when A(t)=5000 (table or calculator).
    – Table shows t=8 is first year where A\ge5000.
    – So investment reaches $5000 during the 8th year.

Radioactive-Decay Example (Carbon-14)

  • Carbon-14 amount modeled by
    C(t)=A\,\bigl(0.999879\bigr)^{t},\quad t=\text{years}.

  • A fossil contains $0.5\,$g after t=50{,}000 yr.
    Set C(50{,}000)=0.5 and solve for the original amount A:
    \displaystyle A=\frac{0.5}{0.999879^{50{,}000}}\approx2.12\,\text{g}.

  • Illustrates exponential decay that happens continuously in nature, not in discrete steps.

Discrete vs Continuous Compounding

  • Banking/finance: compounding happens on discrete schedule (daily, monthly, etc.).

  • Natural processes (radioactive decay, population growth of bacteria, etc.) act continuously.

  • Conceptual leap: let compounding frequency m\to\infty to model “continuous” compounding.

Continuous Compounding & Euler’s Number e

  • Toy experiment: invest $1 for 1 yr at 100 % interest, compounded m times/year:
    A(1)=\left(1+\frac1m\right)^{m}.

  • As m\to\infty, the limit of \bigl(1+1/m\bigr)^{m} approaches e\approx2.718281828\ldots.

  • e (Euler’s number) is the limiting factor for continuous growth; crucial in calculus and finance.

Continuous-Compounding Formula & Syntax

  • For principal P, annual rate r (decimal), time t (years):
    \boxed{\displaystyle A(t)=P\,e^{rt}}.

  • Growth if r>0; decay if r<0.

  • Calculator shortcuts:
    – TI-83/84: e^(...) or EXP(...).
    – Excel: =EXP(rt).

Worked Example 2 – Growth (Bank Account)

  • P=10{,}000\,,\;r=0.06\,,\;\text{continuous}.

  • Balance function: A(t)=10{,}000\,e^{0.06t}.

  • After t=5 yr: A(5)=10{,}000\,e^{0.3}\approx13{,}498.59.

Worked Example 3 – Continuous Decline (Stock)

  • Same P=10{,}000 but declining at 6 % continuously ⇒ r=-0.06.

  • A(t)=10{,}000\,e^{-0.06t}.

  • After 5 yr: A(5)=10{,}000\,e^{-0.3}\approx7{,}408.18.

Introduction to Logarithms

  • Definition: \displaystyle y=\log_{b}(x)\iff b^{y}=x.
    – “The power you raise b to in order to obtain x.”

  • Examples:

    • \log_{2}(8)=3\;\;\because\;2^{3}=8.

    • \log_{3}\bigl( 1/27 \bigr)=-3\;\;\because\;3^{-3}=1/27.

  • Domain restrictions: b>0,\;b\neq1,\;x>0.
    – \log{3}(-9) undefined (cannot reach a negative with positive base). – \log{1}(2) impossible (base 1 always returns 1).

Special Bases

  • Common logarithm: \log_{10}(x), usually written just \log(x).

  • Natural logarithm (Napierian): \log_{e}(x), written \ln(x).

  • Quick evaluations:

    • \log(10{,}000)=4 because 10^{4}=10{,}000.

    • \ln(e)=1\;;\;\ln(1)=0.

Change-of-Base Formula & Calculator Tips

  • \displaystyle \log_{b}(a)=\frac{\log(a)}{\log(b)}=\frac{\ln(a)}{\ln(b)}.

  • Allows any base to be computed on a standard calculator.
    – Example: \log_{11}(9)=\dfrac{\ln(9)}{\ln(11)}\approx0.91631.

Solving Exponential Equations with Logs

  • Equation 5^{-x}=125 ⇒ rewrite as log:
    -x=\log{5}(125) \Rightarrow x=-\log{5}(125)=-\frac{\ln125}{\ln5}=-3.

  • Equation 3^{\;2x-1}=6:
    2x-1=\log{3}(6)\;\Rightarrow\;x=\dfrac{\log{3}(6)+1}{2}\approx1.3155.

Logarithmic Function (General Form)

  • Simplest descriptive form:
    \displaystyle f(x)=a\,\ln(x)+c, \quad a\neq0.

  • More general statement: any f(x)=\log_{b}(x)+c for positive base b\neq1 is “logarithmic.”

  • These will be graphed and analyzed (shifts, stretches, asymptotes) in next lecture.

Ethical, Historical & Practical Notes

  • John Napier (17th century) introduced logarithms to simplify enormous hand calculations; enabled Kepler’s celestial computations.

  • e surfaces naturally whenever growth/decay is continuous, linking finance, physics, biology, chemistry and information theory.

Looking Ahead

  • Upcoming topics:
    – Graphing log functions & identifying asymptotes.
    – Logarithmic identities (product, quotient, power rules).
    – Applications: investment doubling time, radioactive half-life, logarithmic regression.