Redefining Exponentials and Logarithms

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Defining the Natural Logarithm

• Recall the power rule for integrals:

• This means that the power rule does not work when due to a division by zero.
• Instead, we define the natural logarithm by the integral of instead.

Definition
Natural Logarithm

For any , we define the natural logarithm as:

• This definition would exhibit many interesting properties about its value.
  • For : this represents the area under the curve from to .
  • When , we rewrite the following expression as:
  • Therefore, is negative when .
  • Notice that when , we are simply differentiating at a point therefore

Properties of the Natural Logarithm

• By taking this definition of the natural logarithm, we can use the unique definition by essentially proving the properties of the logarithm.

Derivative of the Natural Logarithm

• Due to the way we defined the natural logarithm, we can simply use the Fundamental Theorem of Calculus.

Definition
Derivative of the Natural Logarithm

For , the derivative of the natural logarithm is given by:

Proof

When differentiating either side, we apply the Fundamental Theorem of Calculus.

• By using an absolute value function to create the function , we can extend the domain of the function from to include any real number (except when ).
• Doing this, we can now define the antiderivative of

Definition
Integral of

The natural logarithm is the antiderivative of the function .

Logarithmic Properties

• We review the logarithmic properties of the natural logarithm, then prove them at the same time.

Definition
Properties of the Natural Logarithm

If and is some rational number, then the following properties should hold:

Proving First, we use the definition of the logarithm.

Substituting :

The problem becomes integrating at a point, and area at a point is 0, therefore .

---

Proving We use the definition to express the left-hand side as:

We separate the integral into two separate integrals by integrating between and instead.

u = \frac{t}{a} = \frac{a}{a} = 1

u = \frac{t}{a} = \frac{ab}{a} = b

\begin{align*}
\ln(ab) &= \int_{1}^{a}\frac{1}{t}, dt + \int_{a}^{ab}\frac{1}{t}, dt\
&= \int_{1}^{a}\frac{1}{t}, dt + \int_{a}^{ab}\frac{a}{t}\frac{1}{a}, dt\
&= \int_{1}^{a}\frac{1}{t}, dt + \int_{1}^{b}\frac{1}{u}, du\
\end{align*}

\ln(ab) = \ln a + \ln b

Proving

Differentiating with respect to gives us:

Furthermore, differentiating gives us:

Since the derivatives of these two functions are the same, by the Fundamental Theorem of Calculus, they must differ by a constant. Therefore,

Taking , we get:

Therefore, the proof for is complete.

The Number

• The number is a special number defined from the natural logarithm.

Definition
Euler's Number

The number is defined to be a real number, such that:

• This says that the area under the curve of from to is .
• The values of is given approximately as:

The Exponential Function

• Since the natural logarithm is a one-to-one function, it should have an inverse function.
• We then define its inverse function as the function .

Definition
The Exponential Function

For any real number , we define for which:

• Therefore, due to this definition, we can say that:
  • for all

Properties of the Exponential Function

• By defining as an inverse function, we lose the properties associated to the laws of exponents.
• We can still verify them by expressing them using the inverse definition of the exponential functions.

Definition
Properties of the Exponential Function

If and are any real numbers and is a rational number, then:

Proving We start from .

Since is one-to-one, then:

Derivative of the Exponential Function

• To verify the differentiation formula for , we can do so by knowing that is the inverse of the natural logarithm.
• We can differentiate inverses this by using implicit differentiation.

Proving the Derivative of the Exponential Function

Let .
Since it is defined as the inverse of the natural logarithm, we take the natural logarithm from both sides of the equation.

Taking the derivative on both sides via implicit differentiation, we get:

Since , we see that:

This also verifies the integration formula:

General Exponential and Logarithmic Functions

• We can also define exponential and logarithmic functions with bases other than
• For instance, we can define the general exponential as:

Definition
General Exponential Function

For any and for any real number , we define as:

• We can use this to generalize the properties and for any irrational value of .
• We can also use this to differentiate exponentials with bases other than .

Proving the Differentiation and Integration Formulas for

Let , the differentiate.

---

Let , then set up -substitution.
Let , then .

• We can also use this idea to derive general logarithm functions from the general exponential.
• By knowing that is one-to-one and has a well-defined inverse for , then we can define the general logarithm as:

Definition
General Logarithmic Function

For any real number and , the general logarithmic function is defined as:

• We can use this definition by verifying the change-of-base formula for logarithms.

Proving the Change-of-Base Formula for Logarithms

Let , then express it in terms of .

Then, we take the natural logarithm of both sides.

• We can then use this proof to derive the differentiation formula for a general logarithm with a base .

Proving the Differentiation Formula for the General Logarithm

Let .