Logarithm Form y = log x or y = ln x

        Exponential Form 10^y =x, x> 0 or e^y = x, x > 0, or  b^t = x, t > 0,

        General Rule: y = log x <==> 10 ^y = x

II. Exponential Formula:

        P_(t) = P₀ (1 + r )^t , b > 0 and 1

        (1  r) growth rate, where r is the decimal represent of percent rate of change

        P₀ is the initial value of P_(t) at t = 0

        (1 + r ) > 1 gives exponential growth ---- increasing function

        0 < (1 - r ) < 1 gives exponential decay --- decreasing function

III. Exponential Functions: P_(t) = P₀ (1  r )^t

        (1  r)^t = 

        log (1 r)^t = log ()

        t log (1 r) = log ()

        t = f(log x)

        (1+r) is growth - increasing function

        (1-r)  is decay - decreasing function
 

IV Logarithm Properties: Examples
 

Common Logarithm - to the base 10      y = log x , 10y = x, x > 0 1. log 1 = 0 and log 10 = 1 2. Exponential : log 10x = x, for all x     10log x = x, for x > 0 3. For positive values of a and b:    log(ab) = log a + log b    log (a/b) = log a - log b    log bt = t log b

Natural Logarithm - to the base e        y = ln x , ey = x, x > 0 1. ln 1 = 0  2. Exponential : ln ex = x, for all x      elog x = x, for x > 0 3. For positive values of a and b:      ln(ab) = ln a + ln b      ln (a/b) = ln a - ln b      ln bt = t ln b

V. Continuous Growth Formula:

        P(t) = P₀ e^rt, where P₀ is the initial value, r is the decimal equivalence of the percent rate of change,
        and t is the time period that the rate of growth / decay is  applied.

        When k > 0, P(t) is an increasing function, i.e. Continuous percent growth rate

        When k < 0, P(t) is a decreasing function

VI. Converting Between P(t) = P₀(1+r)^t and P(t) = P₀ e ^kt:

        If (1+r) = e ^k, then ln (1+r) = ln (e^k)

        So k = ln(1+r)

VII. Compound Interest:

        P($) = P₀ (1+)^{n t} , where r is the rate, n is the number of times compounded annually,

        P₀ is the initial investment, and P(t) is the investment at time t in years.

VIII. Continuous Compounding:

        P(t) = P₀ e^{r t}, where r is the annual percent rate in decimal equivalence

IX. Effective annual percent rate:

    Is the % rate equivalent to r, in (1+r) after compounding:  = e ^r = (1+"r")
    where "r" is greater than the nominal rate (constant percent rate, r)