Topic Two (1.2): Logarithmic and Exponential Functions

 Introduction

The Exponential Function, written $\text{exp}\:x$ or $e^x$, is the function whose derivative is equal to its equation. In other words: If $y=e^x,\:\frac{dy}{dx}=e^x$ and if $y=e^{kx},\:\frac{dy}{dx}=ke^{kx}$ where $k$ is a constant.

Because of this special property, the exponential function is very important in mathematics and crops up frequently. Like most functions you are likely to come across, the exponential has an inverse function, which is $\log_ex$, often written $\ln x$. Remember that $e$ is the exponential function, equal to $2.71828...$ On a graph, $e^x$ and $\ln x$ are reflections of one another in the line $y=x$.

Logarithms are another way of writing indices. If $y=a^x$ then $x=\log_ay$.

Laws of Logs: The properties of indices can be used to show that the following rules for logarithms hold:

$\log_ax+\log_ay=\log_a\left(xy\right)$;

$\log_ax-\log_ay=\log_a\left(\frac{x}{y}\right)$; 

and $\log_ax^n=n\:\log_ax$.

The Natural Logarithm: $\ln x$  is also known as the natural logarithm. The derivative of $\ln x$ is $\frac{1}{x}$. It therefore follows that the integral of $\frac{1}{x}$ is $\ln x+c$.

Solving Equations: If you are given equations involving exponentials or the natural logarithm, remember that you can take the exponential of both sides of the equation to get rid of the logarithm or take the natural logarithm of both sides to get rid of the exponential. Logarithms can also be used to help solve equations of the form $a^x=b$ by taking logs of both sides.

 Reading

For this topic, you will need to work through pages 25 to 51 of your textbook.

 Tasks

Complete the following:

• Start by reading through pages 25 to 26 as an introduction to this chapter.

Logarithms to base $10$

• Work through pages 26 to 30, including exercise 2A.

Logarithms to base $a$

• Work through pages 30 to 33, including exercise 2B.

The laws of logarithms

• Work through pages 33 to 35, including exercise 2c, BUT omitting Question 8d.
• The following link provides a nice explanation of these:

WATCH – Rules of Logs (Exam Solutions) Links to an external site.

Solving logarithmic equations

• Work through pages 35 to 40, including exercise 2E.
• If you need extra help, scroll down the following webpage to the second section:

WATCH – Exponential and Log Equations (Exam Solutions) Links to an external site.

Solving exponential equations

• Work through pages 40 to 41, including exercise 2F.
• If you need further help, watch the first section of the video listed above.
• There is a series of 4 video tutorials on this topic:

WATCH – The Exponential Function $e^x$ (Exam Solutions)

WATCH – Sketching Exponential Graphs Based on Transformations (Exam Solutions) Links to an external site.

WATCH – The Natural Logarithmic Function, $\ln x$ (Exam Solutions)

WATCH & REVIEW – Exam Questions: Natural Log Functions (Exam Solutions) Links to an external site.

• The following video also contains great examples on natural logs and exponentials:

WATCH – Basic Intro to the Function $\ln\left(x\right)$ The Natural Log (YouTube)

Natural logarithms

• Work through pages 42 to 43, including exercise 2G.

Transforming a relationship to linear form

• Work through pages 44 to 48, including exercise 2H.

 Consolidate

Now consolidate your understanding by making revision cards for this chapter (the checklist on page 49 will help) and work through the end-of-chapter review on pages 50 to 51.

 Assignment

When you have completed all the activities and are fully prepared and feel confident with the material, you should complete Assignment One and submit it to your tutor via Canvas for marking and feedback.