lecture recording on 18 June 2025 at 21.26.09 PM

Exponential and Logarithmic Functions

Introduction

Logarithmic and exponential functions, like trigonometric functions, are essential for accurate modeling in various fields.
The order of introducing trigonometric and exponential/logarithmic functions is not critical.
The inclusion of these functions alongside polynomial functions enables more accurate representations of real-world phenomena.
This accuracy is crucial when modeling functions to determine rates of change, necessitating models that may incorporate exponential, logarithmic, trigonometric, and polynomial elements.

Exponential Functions

Definition: An exponential function has a constant base and a variable exponent, such as y = 2^x.
In contrast, a polynomial function like y = x^2 has a variable base and a constant exponent.
Exponential functions exhibit rapid growth, especially as the variable in the exponent increases.

Alterations

Functions can be altered through vertical shifts, horizontal shifts, compressions, and stretches.
Understanding the organic form of functions, like x^2 or 15^x, involves recognizing their basic structure before alterations.

Comparison with Polynomials

Comparing y = x^2 and y = 2^x:
- They intersect at (2, 4), meaning they have the same value at that point.
- Exponential functions increase at a faster rate than polynomial functions as x increases beyond a certain point.

Common Point

Functions like y = 100^x and y = 2^x share a common point at (0, 1).
Any number to the power of zero equals one (i.e., a^0 = 1).
This point serves as a ground zero or connected point for unaltered functions.

Effect of Negative Sign

Changing the sign of an exponential function, such as from y = 100^x to y = -100^x, negates the output, causing it to approach negative infinity instead of positive infinity.
Using negative exponent, such as y = 100^{-x}, flips the function, causing it to decrease towards zero as x increases.

Convergence and Divergence

Functions can either converge or diverge:
- Converging: Approaching a specific value or limit.
- Diverging: Moving away and having irregular outputs.

Logarithmic Functions

Logarithmic functions are closely related to exponential functions.
Examples include \log_b(x) and \ln(x).

Relationship with Exponential Functions

Exponential and logarithmic functions are inverses of each other.
For instance, y = 100^x and y = \log_{100}(x) are inverse functions.
Their graphs are symmetric about the line running diagonally through quadrants one and three.

Asymptotes

Exponential functions, such as y = 100^x, have a horizontal asymptote at y = 0.
They do not have a vertical asymptote because they are defined for all real numbers.
Although, it can appear that the graph of an exponential function meets the x-axis, it does not since there is a horizontal asymptote there.

Function Properties

Vertical Line Test: Ensures that a graph represents a function.
- A vertical line drawn at any x value intersects the graph at most once.
Horizontal Line Test: Determines if a function is one-to-one.
- A horizontal line drawn at any y value intersects the graph at most once.
- One-to-one functions have a unique y output for every x input.

One-to-One Property

Exponential and logarithmic functions are one-to-one.
The function y = x^2 is not one-to-one because multiple inputs can produce the same output (e.g., 2 and -2 both yield 4).

Euler's Number

The exponential function e^x is unique and significant.
e is Euler's number, approximately equal to 2.718.
When the base b in an exponential function is Euler's number, denoted as e, it has distinct characteristics.

Logarithmic Forms

General logarithmic function: \log_b(x)
Natural logarithm: \ln(x)
Conversion: \ln(x) = \log_e(x)
Common logarithm: \log_{10}(x); if no base is written, base is assumed to be 10.
For the base b in \log_b(x), b > 0 and b \neq 1.
The argument x in \log_b(x) must be greater than zero (i.e., x > 0).

Domain and Range

For logarithmic functions:
- Domain: All real numbers greater than zero.
- Range: All real numbers.
Logarithmic functions can produce negative outputs.

Practical Implications

When dealing with functions, especially in advanced contexts, it is crucial to verify that the domain is suitable for the intended application.
In many introductory contexts, functions are often presented in a manner that ensures they are always computable, which may not always be the case when working with complex real-world models.

Derivatives of Exponential Functions

Goal: Find the derivatives of exponential and logarithmic functions.

Derivative of e^x

Definition of the derivative: f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
Applying the definition to f(x) = e^x:
\lim_{h \to 0} \frac{e^{x+h} - e^x}{h}
Using the property e^{x+h} = e^x \cdot e^h:
\lim_{h \to 0} \frac{e^x \cdot e^h - e^x}{h}
Factoring out e^x:
\lim_{h \to 0} e^x \frac{e^h - 1}{h}
Applying the constant multiple rule:
e^x \lim_{h \to 0} \frac{e^h - 1}{h}
The limit \lim_{h \to 0} \frac{e^h - 1}{h} = 1
Therefore, the derivative of e^x is: \frac{d}{dx} e^x = e^x

Example

Find the derivative of f(x) = 15e^x
f'(x) = 15e^x, by applying the constant multiple rule.

Significance

The meme: The derivative of e^x is e^x.

Derivatives of Logarithmic Functions

Derivative of \ln(x)

f(x) = \ln(x), where x > 0
The derivative of \ln(x) is: \frac{d}{dx} \ln(x) = \frac{1}{x}

Derivative of \log_b(x)

Converting logarithmic to exponential form:
- \log_b(x) = y \Leftrightarrow b^y = x
Solving for y using natural logarithms:
\ln(b^y) = \ln(x)
y \cdot \ln(b) = \ln(x)
y = \frac{\ln(x)}{\ln(b)}
\ln(b) is a constant.
Taking the derivative:
y' = \frac{1}{\ln(b)} \cdot \frac{1}{x}
y' = \frac{1}{\ln(b) \cdot x}
Thus:
\frac{d}{dx} \log_b(x) = \frac{1}{\ln(b) \cdot x}

Consideration of Quotient Rule

Instead of using the method above, one could also use the quotient rule for y = \frac{\ln(x)}{\ln(b)}, though this method is more complex.

Summary of Logarithmic Derivatives

If f(x) = \log_b(x), then f'(x) = \frac{1}{\ln(b) \cdot x}

Derivative of General Exponential Function

If f(x) = b^x, then f'(x) = b^x \cdot \ln(b)

Summary of Derivative Rules

\frac{d}{dx} e^x = e^x
\frac{d}{dx} \ln(x) = \frac{1}{x}
\frac{d}{dx} \log_b(x) = \frac{1}{\ln(b) \cdot x}
\frac{d}{dx} b^x = b^x \cdot \ln(b)

General Properties

Taking the derivative of exponential functions results in a copy of the original function, which simplifies differentiation.

Examples

Example 1

Find the derivative of f(x) = 2x^2 - e^x + \log_{17}(x)
Applying the different sum rule:
f'(x) = 4x - e^x + \frac{1}{\ln(17) \cdot x}

Example 2

Find the derivative of f(x) = \frac{\ln(x)}{e^x + 15x}

Using the Quotient Rule

Applying the quotient rule: \frac{d}{dx} \frac{u}{v} = \frac{u'v - uv'}{v^2}
f'(x) = \frac{\frac{1}{x} (e^x + 15x) - \ln(x) (e^x + 15)}{(e^x + 15x)^2}
Simplified:
f'(x) = \frac{\frac{e^x + 15x}{x} - \ln(x) (e^x + 15)}{(e^x + 15x)^2}

Using the Product Rule

Rewriting as a product: f(x) = \ln(x) \cdot (e^x + 15x)^{-1}
Applying the product rule and chain rule:
f'(x) = \frac{1}{x} \cdot (e^x + 15x)^{-1} + \ln(x) \cdot (-1)(e^x + 15x)^{-2} \cdot (e^x + 15)
Simplified:
f'(x) = \frac{1}{x (e^x + 15x)} - \frac{(e^x + 15) \ln(x)}{(e^x + 15x)^2}

Product Rule vs Quotient Rule

Both rules are correct if computations and simplifications are correct.
Some may prefer using one rule or the other.

Example 3

Calculate the derivative of: \frac{d}{dx} (17^x \cdot e^x)
Using the product rule: f'(x) = (17^x \cdot \ln(17)) e^x + e^x (17^x)
Removing common factors:
f'(x) = 17^x e^x (\ln(17) + 1)

Chain Rule with Exponential and Logarithmic Functions

The chain rule is vital when dealing with composite functions.
If y = f(g(x)), then \frac{dy}{dx} = f'(g(x)) \cdot g'(x)

Derivative of e^{g(x)}

\frac{d}{dx} e^{g(x)} = e^{g(x)} \cdot g'(x)

Derivative of \ln(g(x))

\frac{d}{dx} \ln(g(x)) = \frac{1}{g(x)} \cdot g'(x)

Derivative of \log_b(g(x))

\frac{d}{dx} \log_b(g(x)) = \frac{1}{\ln(b) \cdot g(x)} \cdot g'(x)

Derivative of b^{g(x)}

\frac{d}{dx} b^{g(x)} = b^{g(x)} \cdot \ln(b) \cdot g'(x)

Examples of the Chain Rule

Example 1

Calculate the derivative of \ln(15x^2)
\frac{d}{dx} \ln(15x^2) = \frac{1}{15x^2} \cdot 30x = \frac{30x}{15x^2}

Example 2

Find the derivative of \ln(6x - 12 + e^x)
\frac{d}{dx} \ln(6x - 12 + e^x) = \frac{1}{6x - 12 + e^x} \cdot (6 + e^x) = \frac{6 + e^x}{6x - 12 + e^x}

Importance of Writing as a Single Quotient

Often, problems require the answer to be expressed as a single quotient.

Example 3

Find the derivative of e^{2x^2 - \ln(3)}
\frac{d}{dx} e^{2x^2 - \ln(3)} = e^{2x^2 - \ln(3)} \cdot (4x) + 0 = 4xe^{2x^2 - \ln(3)}
Recognize constants even if they are not the typical rational numbers such as 1, 2, etc.

Derivative of logarithm when there is no variable.

Make sure to recognize constants such as \ln(3) because its derivative is considered zero.

Example 4

Find the derivative of the base function of \log_{2.5}(e^{-sin(2x)} + 13^{cos(x)})
\frac{d}{dx}log_{2.5}(e^{-sin(2x)} + 13^{cos(x)}) = \frac{e^{-sin(2x)} (-cos(2x) * 2 + \ln(13) * 13^{cos(x)} (-sin(x)) } }{\ln(2.5) ( e^{-sin(2x) + 13^{cos(x)} ) } }

Combination of Multiple Functions

Derivative Rules and Chain Link

Applying derivatives follows after knowing specific derivatives such as for the derivative of cosine, logarithm, exponential, and polynomial.
Also apply derivatives to any new functions.

Example 1

Find the derivative of \ln(7x) \cdot b^{\tan(x)}
\frac{d}{dx} [ ln(7x) b^{tan(x)} ] = \frac{1}{7x} * 7 b^{tan(x)} + ln(7x) bln(b)sec^{2}(x)

Things to Avoid

Don't multiply non compatible numbers such as when having a rational number as the base within an exponential such as 5 to the power of something with a rational and then multiplying it by 3, don't multiply it to 15 since it's the base will change the function.

Example 2

\frac{d}{dx} [ 5^{3x} ] = 5^{3x} ln(5) * ( 3)

Key Points

Combination of Multiple Functions
Chain Rule and Derivative Rules for Exponential and Logarithmic Functions , and Trigonometric Functions