## 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 Δ*y*/Δ*t*, provided the changes Δ*y* and Δ*t* 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 Δ*y*/Δ*t* = *ky
*

And it’s worth recalling;

*a*^{x}a^{y}= a^{x+y}

*(a*^{x})^{y}= a^{xy}

*a*^{-x}= 1/a^{x}

*a*^{x/y}=(a^{1/y})^{x}

If *y*(*x*) = *e ^{x}* then the rate of change Δ

*y*/Δ

*t = y(x)*

The constant *e* is an irrational number ≈ 2.718, and the function *e ^{x}* 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 Δ*y* /Δ*x* = *ky(x)*.

Or any exponential change, i.e. any change in which Δ*y*/Δ*x* =* ky*,

*y(x) = y _{0} exp(kx)
*

Where *y _{0}* 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* = 10^{y} i.e.

*y* = log_{10}(*x*)

In other words, log_{10}(*x*) is the power to which we must raise 10 to obtain *x*.

10^{log10(x)} = *x* and log_{10}(10^{x}) = *x*

The **logarithm to base a** of

*x*is the number

*y*such that

*x = a*

^{y}. 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*=

*a*

^{y}then

*y*= log_{a}(*x*)

if *a*^{loga(x) }then log_{a}(*a*^{x})=*x*

The logarithm to base e of *x* is the number *y* such that *x* = *e*^{y}, i.e. if *x* = *e*^{y} then *y* = log_{e}(*x*)

And hence *exp*[log_{e}(*x*)] = *x* and log_{e}[*exp*(*x*)] = *x*

If *y* = *y*_{0}

*exp*(*kt*) then log_{e}(*y*/*y*_{0}) = *kt* and *t* = (1/*k*)log_{e}(*y*/*y*_{0})

**Properties:
**

log_{a}(*xy*) = log_{a}(*x*) + log_{a}(*y*)

log=(*x/y*) = log_{a}(*x*) – log_{a}(*y*)

log_{a}(1/*x*) = -log_{a}(*x*)

log_{a}(*x*^{b}) = *b* log_{a}(*x*)

log_{a}(*x*) log_{b}(*a*) = log_{b}(*x*)

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