Modelling Curves using Logarithms - Revision Notes

In these revision notes for Modelling Curves using Logarithms, we cover the following key points:

• What are functions?
• How do we recognize a function?
• What are the names of the variables in a function? Why are they called so?
• What is the graph of a function?
• What is the exponential function? How to find its components?
• How do we plot the graph of an exponential function?
• How do we plot the graph of logarithmic functions.
• How do we find the original function when its graph is given?
• What does "modelling a curve" mean?
• Why is modelling a curve useful when studying exponential functions?
• How do we make the curves' modelling by using logarithms?

Modelling Curves using Logarithms Revision Notes

A function is a one-to-one relationship between each term of the independent variables set (usually denoted by x) to the corresponding terms of the dependent variables set (usually denoted by y). This one-to-one relationship is often given by a formula but in many cases, it is shown graphically through the graph's line.

An element of the dependent variables' set may have two or more different values in correspondence from the independent variables' set and this relationship is still a function, but the reverse is not allowed in functions. In simpler words, a y-value may have two or more x-values in correspondence but the reverse cannot happen, or if this happens, that kind of relationship between the x- and y-values is not a function.

The graph of a function shows the line that contains all the possible combinations between the x-values and the corresponding y-values of that function. Depending on the formula of the function, the graph may be a straight line (in linear functions), a parabola (in quadratic functions, a hyperbola (in the inverse functions, i.e. in functions of the form y = k/x) and so on. Exponential and logarithmic functions have their corresponding graphs too.

A good method to check whether a given graph represents a function or not is to draw a vertical line in the sections where you have doubts that an x-value may have two or more y-values in correspondence. If the vertical line intercepts the graph at more than one point, that graph does not represent a function.

By definition, an exponential function is a function that has the independent variable in the exponent of one of its terms. The simplest general form of exponential functions is

where 'a' is a constant. More advanced exponential functions include the following forms

where a, b, m and n are all numbers.

The graph of an exponential function is a curve that extends to infinity on three sides but has some limitations on the other, depending on the value of the base and the sign of the exponent. The best method to plot the graph of an exponential function is to find as many points of the graph as possible. If the formula of the function is known, we can give some arbitrary values to the independent variable x and then calculate the corresponding y-values.

Based on the above two criteria (base and sign of exponent), we can list the following cases about the exponential function graphs.

1. The base is greater than 1 and the coefficient m before the variable x in the exponent is positive (a > 1 and m > 0).
   The graph of such functions extends above zero in the vertical axis (y-axis) to infinity. On the other hand, it extends to the infinity on both sides of the horizontal axis (x-axis) but at a faster rate on the left, where it approaches zero. Moreover, the graph is decreasing when moving from left to right.
   This form also includes graphs of the exponential functions such as
   y(x) = a^mx; y(x) = a^{mx + n}; y(x) = b ∙ a^x; y(x) = b ∙ a^mx and y(x) = b ∙ a^{mx + n}
   with the condition that m > 0, n > 0 and b > 1. The only difference is that the graph may be more stretched or compressed laterally, depending on the values of a, b, m and n.
2. The base is greater than 1 and the coefficient m before the variable x in the exponent is negative (a > 1 and m < 0).
   The graph of such functions extends above zero in the vertical axis (y-axis) to infinity. On the other hand, it extends to the infinity on both sides of the horizontal axis (x-axis) but at a faster rate on the right, where it approaches zero. Moreover, the graph is decreasing when moving from left to right.
   This form also includes graphs of the exponential functions such as
   y(x) = a^mx; y(x) = a^{mx + n}; y(x) = b ∙ a^x; y(x) = b ∙ a^mx and y(x) = b ∙ a^{mx + n}
   with the condition that m < 0, n < 0 and b > 1. The only difference is that the graph may be more stretched or compressed laterally, depending on the values of a, b, m and n.
3. The base is smaller than 1 (but greater than 0 anyway) and the coefficient m before the variable x in the exponent is positive (a < 1 and m > 0).
   This form also includes graphs of the exponential functions such as
   y(x) = a^mx; y(x) = a^{mx + n}; y(x) = b ∙ a^x; y(x) = b ∙ a^mx and y(x) = b ∙ a^{mx + n}
   with the condition that m > 0, n > 0 and b < 1. The only difference is that the graph may be more stretched or compressed laterally, depending on the values of a, b, m and n.
4. Likewise, the graph of the exponential function y(x) = a^-x when a < 1 is similar to y(x) = a^x when a > 1 (discussed in the first case).
   This form also includes graphs of the exponential functions such as
   y(x) = a^mx; y(x) = a^{mx + n}; y(x) = b ∙ a^x; y(x) = b ∙ a^mx and y(x) = b ∙ a^{mx + n}
   with the condition that m < 0, n < 0 and b < 1. The only difference is that the graph may be more stretched or compressed laterally, depending on the values of a, b, m and n.

If a < 0, the graph is inverted down with respect to the graph of the corresponding exponential function with a > 0.

We use the following procedure to find the formula of an exponential function of the type y(x) = b · a^x when

Step 1: Find the y-intercept, as this point helps identify the value of the constant b. This is because the y-intercept has the horizontal coordinate x = 0.

Step 2: Choose the coordinates of a known point of the graph to substitute in the general formula of the function in order to find the value of the base a. In this way, the formula of the unknown exponential function is already found.

A logarithmic function is a function that has the variable in the argument of a logarithm. The simplest logarithmic function is

More advanced forms of logarithmic functions include

The graph of a logarithmic function (equation) shows the line that includes all points that make the equation true.

The graph of a logarithmic function is a curve that extends to infinity on three sides but has some limitations on the other, depending on the value of the base and the sign of the exponent. For example, the above graph is unlimited in both vertical directions and on the right but it is limited on the left (x cannot be zero or negative).

The best method to plot the graph of an exponential or logarithmic function is to find as many points of the graph as possible. If the formula of the function is known, we can give some arbitrary values to the independent variable x and then calculate the corresponding y-values.

We have four types of numbers (coefficients or constants) that affect the shape and orientation of a logarithmic function graph: a, b, m and n, where their values and signs determine the orientation and dimensions of the graph, as in the exponential functions.

We can find the values of the coefficients involved in the logarithmic function y(x) = m/n log_ax by solving the simplest form of the logarithmic obtained when substituting the coordinates of one known point of the graph. This allows us to find two unknowns at once: the base a and the coefficient m/n preceding the logarithm. Then, if necessary, we can make some attempts to find the values of m and n separately, given that they are whole numbers and the number of possible combinations is therefore limited.

Modelling curves using logs means transforming curves into straight lines, which not only helps find new values besides those collected by the experiment but also to find the curve's equations if it is missing.

There are two types of functions that require curve modelling using logarithms:

1. Functions of the form y = k · xⁿ

The process of modelling curves expression by the equation above requires the original function to be written in logarithmic form. This means that if we have the original function in the form

where k and n are coefficients (numbers) and x is the independent variable, we must take the logarithm of both sides to obtain

Given that k is a constant, so will log k be as well. Therefore, we obtain a form of this function that is very similar to the formula of a linear equation

where the coefficient m represents the gradient of the line and n is the constant of the equation (here, of the linear function).

We model the curve by expressing on the horizontal axis not the x-values but the log x ones instead. Likewise, on the y-axis, we now show the log y values instead of the y-ones. In such functions, the coefficient n of the logarithmic function acts as a gradient for the straight line after modelling the curve.

Functions of the form y(x) = a · b^x

These functions are considered as pure exponential functions becuase the independent variable is in the exponent. The transformation made in such functions for modelling them is

This form is similar to that of a linear function, where log b is the gradient and log a is the constant of the line. Therefore, this time we have to plot the log y vs x graph instead of log y vs log x one. This means that only the values in the vertical axis need to be transformed.

The procedure used to find the missing coefficients is as follows:

Step 1: First, we determine the constant log a. This represents the value of log y for x = 0. In other words, it represents the initial vertical coordinate of the line, i.e. log a = log (0).

Step 2: Then, we calculate the gradient log b of the line using the relation

Step 3: Then, we find the coefficient b by the formula

To determine what kind of exponential relationship (if any) as set of data belongs to, you must take a look at the first two or three columns of the table and see whether there is any kind of geometric progression in the values. In this case, the function most probably has the form y(x) = b · a^x. If you are not able to identify such a relationship that resembles a geometric sequence but the values look more like forming an arithmetic one (albeit not exactly), then try the first option, i.e. y(x) = k · xⁿ.