# BM4.3 – The Natural Exponential and Logarithm Functions

Chapter 4, Lesson 3

In this lesson students will:

• understand the relationship between exponentiation and logarithms
• use the natural logarithmic and exponential functions to evaluate expressions
• be introduced to two special properties of the natural logarithm that will be used in upcoming lessons

## Exponentiation & Logarithms

#### The Relationship between Exponentiation and Logarithms

Suppose that we wanted to solve the following three problems for $x$:

1. $2^{x} = 8$
2. $2^{x} = 16$
3. $2^{x} = 10$

For the first, perhaps we might guess $x=3$ since we know $2^{3} = 8$. Similarly for the second, we see that $2^{4} = 16$, so $x=4$ must be the solution. However, we run into a bit of a situation for the third, since there is no whole number that will solve $2^{x}=10$. If you are keen, you might notice that since the right side of $2^{x}=10$ falls somewhere between $2^{x}=8$ and $2^{x}=16$, we must have an $x$-value that lies between three and four. But how exactly do we get to the $x$-value that we need?

The solution lies in the operation known as the logarithm. A logarithm helps us solve an equation of the form $a^{x}=b$. Mathematically, $a^{x} = b$ is equivalent to $x = \log_{a}(b)$, where $a$ is called the base of the logarithm, and $b$ is the input to the logarithm. Essentially, we can think of a logarithm as the opposite operation to exponentiation (similar to how subtraction is the opposite operation to addition).

Revisiting our questions from above, we have

1. $2^{x} = 8$ is equivalent to $x = \log_{2}(8)$, and inputting this into a scientific calculator gives $x=3$.
2. $2^{x} = 16$ is equivalent to $x = \log_{2}(16)$, and inputting this into a scientific calculator gives $x=4$.
3. $2^{x} = 10$ is equivalent to $x = \log_{2}(10)$, and inputting this into a scientific calculator gives $x\approx 3.3219$, so indeed the decimal approximation we were looking for was between three and four!

#### The Natural Logarithmic & Exponential Functions

Unfortunately, our calculator does not have the ability to define logarithms of different bases. It does, however, have one important logarithm and we will focus our attention on this. Along the left side of the BAII Plus calculator, you will see the button. This is a special logarithm known as the natural logarithm, and is sometimes pronounced like “lawn” or spelled out “ell – enn (L – N)” when referring to it. This logarithm has a base of $e \approx 2.7182$, which is referred to as Euler’s number. Euler’s number $e$ is an irrational number, and we will explore how it comes up naturally in a compound interest problem in Lesson 4.10. The concepts of the natural logarithm and exponential are connected in the following way

$e^{x} = b \textrm{ is equivalent to } x = \ln(b).$

Above the natural logarithm button, you will notice as the second function. This function is known as the natural exponential function. Using the button gives you access to this function should we need to use it.

Let’s spend some time learning how to evaluate expressions using the natural logarithm and exponential functions. Suppose we want to calculate $\ln(2)$ or $e^{-3}$. To calculate $\ln(2)$ we first type $2$ into the BAII Plus calculator, then we hit the button. The final answer should be roughly $0.6931$. For $e^{-3}$ we first input $-3$ by hitting the $3$ button, followed by the button. Then we go into the second function menu by clicking . Finally, hitting the button should give a final answer of about $0.0498$.

Give the following few questions a try (note: some may give errors in your calculator). Answers will be supplied in the video below.

1. $\ln(-1)$, $\ln(0)$, and $\ln(1)$
2. $\ln(1.01)$, $\ln(1.001)$, and $\ln(1.0001)$
3. $\ln(5^{3})$ and $3 \times \ln(5)$
4. $e^{-1}$, $e^{0}$, and $e^{1}$
5. $e^{2} \times e^{3}$ and $e^{5}$

#### Special Properties of the Natural Logarithm

There are two special properties of the natural logarithm that we saw in the previous examples and video that I wish to highlight to end this lesson. The first property shown in Example 2 is that, when $i$ is very small and positive, we have

$\ln(1 + i) \approx i.$

We will use this fact when discussing the Rule of 72 in Lesson 4.8. The second special property shown in Example 3 allows us to rewrite a power inside the natural logarithm as a multiplication outside of the natural logarithm. This will be very helpful when trying to isolate a power. In Lesson 4.8 we will see how we can use this fact to help us determine the term of an investment that earns compound interest by using

$\ln(1+i)^{n} = n \times ln(1+i)$

and solving for $n$.