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