W5_Exponential and logarithmic functions

Session Overview

Title: Fundamentals of Mathematics 1

Sessions: 5 & 6

Topics: Exponential and Logarithmic Functions

Instructor: Erwan Lamy

Institution: ESCP BUSINESS SCHOOL

Location: Berlin, London, Madrid, Paris, Turin, Warsaw

Course Objectives

• To introduce exponential functions and their applications.

• To introduce logarithmic functions and their graphs.

• To study the basic properties of logarithmic functions.

• To develop techniques for solving logarithmic and exponential equations.

Outline

• Exponential Functions (Chapter 4.1)

• Logarithmic Functions (Chapters 4.2 and 4.3)

• Logarithmic and Exponential Equations (Chapter 4.4)

Exponential Functions (4.1)

Definition:

The function defined by ( f(x) = b^x ) where ( b > 0 ) and ( b ≠ 1 ) is called an exponential function.

Base:

The base ( b ) can be any positive real number except 1, and the exponent can be any real number.

Rules for Exponents:

• ( b^x × b^y = b^{x+y} )

• ( b^{-x} = \frac{1}{b^x} )

• ( b^1 = b )

• ( (b^c)^d = b^{cd} )

• ( b^0 = 1 )

• ( b^{x} × b^{-y} = \frac{b^{x}}{b^{y}} )

Example: Bacterial Growth Function:

( N(t) = 300 × (4/3)^t )

• Initial bacteria: ( N(0) = 300 )

• Bacteria after 3 min: ( N(3) = 300 × (4/3)^3 \approx 711 )

Methods for Exponential Functions:

• Graphing techniques for visualizing growth using software tools.

• Application of real-life scenarios such as population dynamics and finance.

Graphs of Exponential Functions

• When ( b > 1 ): Graph rises from left to right.

• When ( 0 < b < 1 ): Graph falls from left to right.

Domain and Range:

• Domain: All real numbers.

• Range: Positive numbers only.

Intercepts:

• y-Intercept: (0, 1); x-Intercept: None.

Example: Population Growth Model:

• Population grows at 2% per year.

• Calculation: For initial population 10,000, At ( t = 3 ): ( P(3) = 10000(1 + 0.02)^3 \approx 10612 ).

Example: Compound Interest Formula:

• ( S = P(1 + r)^n )

• Calculation: For principal of $1000 at 3% for 30 years, ( S = 1000(1 + 0.03)^{30} \approx 2427.26 ).

Natural Exponential Function

• Definition: Exponential function with base ( e ), where ( e \approx 2.71828 ).

• Properties: Appears in natural and social phenomena; equal to its own derivative.

Logarithmic Functions (4.2 & 4.3)

Definition:

Logarithmic function of base ( b ) is the inverse of the exponential function of base ( b ). If ( y = b^x ), then ( log_b(y) = x ).

Common and Natural Logarithms:

• Natural: ( ln(x) = log_e(x) )

• Common: ( log(x) = log_{10}(x) )

Properties of Logarithms:

• ( log_b(xy) = log_b(x) + log_b(y) )

• ( log_b\left(\frac{x}{y}\right) = log_b(x) - log_b(y) )

• ( log_b(x^k) = k \cdot log_b(x) )

• ( log_b(1) = 0 )

• ( log_b(b) = 1 )

Change of Base Formula:

( log_a(b) = \frac{log_c(b)}{log_c(a)} )

Example of Logarithmic Computations:

• Calculate: ( log_{10}100 = 2 )

• ( ln(1) = 0 )

• ( log_{10}(0.1) = -1 )

Methods for Logarithmic Functions:

• Utilization of logarithmic properties for simplifying expressions and solving equations.

• Applications in scientific data analysis and growth rate comparisons.

Exponential and Logarithmic Equations (4.4)

Definition:

Logarithmic equations involve logarithms of unknown expressions. Exponential equations have unknowns in exponents.

Example of Solving an Exponential Equation:

• Solve the equation ( 5 = 2^x ), using logarithms: ( x = log_2(5) )

Real-World Application Example:

• Study on Oxygen Consumption: Based on weight, involving logarithmic relationships: Solve ( log(y) = log(5.34) + 0.885log(x) ) for ( y ).

Combined into:

• ( y = 5.34 * 0.885^x ).