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 ).