Exponential and Logarithmic Functions

Question 1

The exponential function f(x) = bˣ, b > 0

Answer 1

Also known as the growth and decay curve this function is neither odd nor even, f(x) crosses the y-axis at (0,1) and y = 0 is a horizontal asymptote (as x→∞, y→0). Domain: x ∈ real numbers, Range: y > 0

Question 2

The exponential function f(x) = eˣ

Answer 2

The number e is irrational and has an important calculus property, it differentiates to itself, this means that at any point on the graph the gradient of the curve at that point is equal to the value of the function (y value) at that point

Question 3

The logarithmic function f(x) = logᵦ(x)

Answer 3

The logarithmic function is the reverse of the exponential function f(x) = bˣ, so the 2 curves are mirror images of each other in the line y = x

The inverse of f(x) = eˣ , is…
f(x) = logₑ(x) = ln(x)
ln(x) is called a natural logarithm

This function is neither odd nor even, f(x) crosses the x-axis at (1,0) and x = 0 is a vertical asymptote (as y→∞ , x→0). Domain: x > 0, Range: y ∈ real numbers

Logarithms are indices of a fixed base number, this base number is usually e = 2.72 or 10

Question 4

Logarithmic notation

Answer 4

If y = bˣ ⇔ x = logᵦ(y), where x ∈ real numbers and y ∈ real numbers, y > 0

b is called base
x is called logarithm
y is the number

Note:
The logarithm of a negative number or zero is not defined

Since the point (0,1) lies on y = bˣ, its reflection (1,0) lies on y = logᵦ(x) for every base b.

When 0 < x < 1, log is negative

Question 5

Properties of logarithmic functions

Answer 5

logᵦ(b) = 1
logᵦ(1) = 0
logᵦ(bⁿ) = n
logᵦ(xⁿ) = n * logᵦ(x)
logᵦ(ⁿ√x) = [1/n] * logᵦ(x)
logᵦ(uv) = logᵦ(u) + logᵦ(v)
logᵦ(u/v) = logᵦ(u) - logᵦ(v)

Question 6

Special bases

Answer 6

Although the base of the logarithm function can be any real positive number except 1, only 2 bases are in common use. One is a number denoted by e, logarithms to base e are denoted to ln (logₑ ⇔ ln() ), the other is base 10 which is important because our system of writing numbers is based on powers of 10. If no base is specified the symbol log stands for log₁₀

Question 7

Equations and inequalities

Answer 7

Change the logarithmic function to exponential function (x = logᵦy ⇔ y = bˣ).
Then taking logs of the expression on both sides of an index equation undoes the effect of the index.

Note:
logₑ(x) = ln(x) on a calculator

Inequalities can be solved in the same way as equations, the inequality sign is reversed when multiplying or dividing by a negative number,
ln(0.8) is negative so the sign must be reversed when you divide or multiply ln(0.8) on both sides.

Question 8

Modelling

Answer 8

Modelling is the name given to the process where we try to describe a natural process in terms of a mathematical formula, such formulae are hardly ever perfect or exact but nevertheless provide a way to roughly describe what is happening in real life

Question 9

Power law model

Answer 9

Here we describe some kind of growth relationship in terms of an exponential equation y = axⁿ, this can be investigated by taking logs for reasons;
1. confirm that a set of data points fits this model
2. calculate the values of the constants a and n, enabling us to describe the relationship in terms of a mathematical equation

This process is done by taking by taking logarithms on both sides of the equation to match the form y = mx + c

We can then find information such as the y-intercept and gradient.

Question 10

Determining the power law model

Answer 10

1. Transform equation by taking logs on both sides
2. Substitute (x₁,y₁) and (x₂,y₂) from 2 suitable points into the transformed log equation
3. Solve the simultaneous equations to find n
4. Substitute (x₁,y₁) and n into y = axⁿ to find a value.