Exponential and Logarithmic Functions

In an exponential change the rate of change of some quantity y, at any time t, is directly proportional to the value of y itself at that time, rate of change of y(t)
y(t) = ky(t)

where k is the constant of proportionality.

If k is positive, y increases with time (Exponential Growth). If k is negative, y decreases with time (Exponential Decay).

For any quantity y that changes at a constant rate, the graph of y against time t is a straight line with a gradient equal to the rate of change of y.

The rate of change of any quantity y, at a particular time, can be represented by the quotient Δyt, provided the changes Δy and Δt are sufficiently small. The value of such a rate of change is given by the gradient of the tangent to the graph of y against t at the time in question.

In an exponential change, at any time Δyt = ky

And it’s worth recalling;

  • axay = ax+y
  • (ax)y = axy
  • a-x = 1/ax
  • ax/y =(a1/y)x

If y(x) = ex then the rate of change Δyt = y(x)

The constant e is an irrational number ≈ 2.718, and the function ex is known as the exponential function and is often written exp(x).

For any function y(x) = exp(kx), the gradient at any value of x is ky(x), and so the rate of change Δyx = ky(x).

Or any exponential change, i.e. any change in which Δyx = ky,

y(x) = y0 exp(kx)

Where y0 is the value of y when x = 0

Common Logarithm

The logarithm to base 10 of x is the number y which satisfies the equation x = 10y i.e.

y = log10(x)

In other words, log10(x) is the power to which we must raise 10 to obtain x.

10log10(x) = x and log10(10x) = x

The logarithm to base a of x is the number y such that x = ay. In other words, the logarithm to base a of x is the power to which we must raise a to obtain x, i.e. if x = ay then

y= loga(x)

if aloga(x) then loga(ax)=x

The logarithm to base e of x is the number y such that x = ey, i.e. if x = ey then y = loge(x)

And hence exp[loge(x)] = x and loge[exp(x)] = x

If y = y0
exp(kt) then loge(y/y0) = kt and t = (1/k)loge(y/y0)

Properties:

loga(xy) = loga(x) + loga(y)

log=(x/y) = loga(x) – loga(y)

loga(1/x) = -loga(x)

loga(xb) = b loga(x)

loga(x) logb(a) = logb(x)

If y = kxp then loga(y) = p loga(x) + loga(k), so a graph of loga(y) against loga(x) has gradient p and intercept loga(k). The logs may have any base, although bases 10 and e are generally used.

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~ by jamesdow2013 on March 29, 2013.

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