Exponential and log functions

Exponential and log functions
This chapter explores exponential and log functions. It covers sketching simple transformations of the graph of y=e^x, sketching simple transformations of the graph y=lnx, solving equations involving e^x and lnx, understanding the definition of the terms exponential growth and decay, and solve real life examples of exponential growth and decay.
Exponential functions are in the form of y=a^x. These type of graphs all pass through the coordinate (0, 1), that’s because a⁰ = 1 for any number in place of a. In the following examples we shall explore sketching exponential graphs.

Example

In this example we shall sketch the graph of f(x) = 2^x.
The first logical step is to draw a table such as that shown below. Chose a good range of x values include a few negative values.

Now we can sketch the graph as shown below.
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All exponential graphs exhibit the same patterns. The standard graph of y=e^x can be used to represent exponential growth, applications of exponential growth include population growth.
The exponential function y=e^x where e=2.718 is identical to its gradient function. For this reason it is referred to as the exponential function.
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Below are some examples of graphs involving e^x

Example

First we shall sketch the graph of y=e^x. Drawing the following table should make the entire process easier. It should also be obvious how the curve grows rapidily.

Below is the curve that is sketched from the table above;
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From the data and sketched curve it should be very clear that;
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The second curve that we will attempt in this example is; y=e^-x. We can draw the table as shown above and then sketch the curve;
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The first curve (y=e^x) and the curve y=e^-x are very similar except curves are reflection of each other as shown below;
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The example is often referred to as exponential decay.
Below are further examples of exponential functions graphs;

Example

First we shall start with y=e^2x. Since it is just a sketch to get an idea of how the curve looks like, you only need to calculate a few values as shown in the table below;

The graph below shows the standard curve of y=e^x and the curve of y=e^2x to show the difference between the two
You might notice that the y values of y=e^2x are the ‘square’ of the y values of y=e^x.

Example

Next we shall try the curve of y=10e^-x. Again we don’t need to calculate all the x values, just a few values to get an idea of the shape of the curve; Below is the table you could make;

Below the curves for y=e^-x and y=10e^-x are 10 times bigger than the y values of y=e^-x

Example

Next we shall try the curve of y=3+4e^½x. Again we just need a few values such as those shown below;

Below is the sketch of the curve in subject;
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Notice that the range of the function is y>3.

Example

In this example we shall use exponential functions in an application. Here is the example;
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The price of the car over a period of time can be represented by the following formula;
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…where P is the price of the car In £s and t is the age in years from its new condition.
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The following examples explore the inverse of the exponential function. Inverse functions perform the ‘opposite’ operation to a function i.e ‘+4’ becomes ‘-4’ and ‘x²’ becomes ‘√x’
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…also…
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The inverse of e^x is often written as lnx. Below are some examples;

Example

Let use start with x² = 10. Suppose we wanted to find x. x is simply √10.
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Next we shall work with e^x – 3. We know the inverse of e^x.
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We can also inverse lnx i.e
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In the examples above we have seen that the inverse of a function is simply the opposite. The graph of lnx is simply the reflection of e^x in the line y=x as shown below;
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For the lnx graph you’ll notice that;
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Let us look at some examples of lnx graphs below;

Example

The first example is the graph of ln(3 -x). The graph has been sketched below;
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…from the graph above you’ll notice that when x→3, y → -∞. There are no y values for values of x larger than 3. …when x=2, y = 0 and as x→-∞, y→∞.
Next we shall try the curve of y=3+ln(2x). Using the same steps as we have been using above, the curve will look like the curve shown below;
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…from the curve above we can see that when x→0, y→-∞ and as x→∞, y→∞ but very slowly.
Next we shall look at solving equations involving exponential and log functions.

Example

Here is an example that looks at solving two equations. These are;
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Let us look at the first equation e^{2x + 3} = 4. Here we shall take advantage of the inverse of e^x which we know is lnx. Use the inverse to get the following;
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We can now simplify the expression and rearrange it to find x;
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Notice that;
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So now we have;
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Now we shall try the next equation that is;
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…we’re trying to find the value of x.
We rearrange to get lnx on its own to get;
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…therefore;
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Example

In this example we shall look at an application. The following represents the number of lions in a single herd.
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…where N stands for the number of lions in the herd and t is the time in years.
We may want to observer how the number of lions changes over the years. Finding the original number is easy, we simply replace t with 0 in the equation as shown below;
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How about after 4 years, again we simply replace t with 4 to get;
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We may want to predict how many years for the population to reach a certain number, for instance how many years it will take to reach 25 lions. We simply replace N with 25 in the equation as shown below;
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Now we can rearrange the expression to find t which stands for number of years.
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Notice that the inverse of e^x is lnx therefore…
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So the population will be over 25 after 28 years.
We can predict the long term model of the population by using a graph and the data we have so far, we have observed that;
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…we can also sketch a graph to observe the exponential as shown below;
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