Covering logic and exponential functions in higher mathematics.
Includes past paper questions and transformations of logarithmic and exponential graphs.
Sponsored by an educational publisher offering a discount on practice question books.
Exponential Functions
Definition: Function of the form ( f(x) = a^x ), where ( a > 0 ) and ( x ) is a real number.
Base ( a ) raised to the power ( x ).
Graph features:
Passes through ( (0,1) ) since ( a^0 = 1 ).
Passes through ( (1,a) ) since ( a^1 = a ).
Graph shape becomes steeper as ( x ) increases.
Asymptotically approaches zero on the left.
Negative values of ( a ) invert the graph.
Graph Examples
Sketching an example: ( y = 6^x )
Passes through ( (0,1) ) and ( (1,6) ).
The Constant ( e )
( e ) is an irrational number ( \approx 2.718 ).
Natural exponential function: ( f(x) = e^x ).
Graph Transformations
Types of transformations:
Vertical shifts, scaling, reflections, and compressions.
Rules for transformations:
Vertical shifts: add/subtract a constant.
Scaling: multiply/divide by a constant.
Reflections: negative sign inverts the graph.
Horizontal shifts: changes inside the function bracket.
Example of Transformation
Example: ( y = 1 + 2^x )
Transforms basic graph ( y = 2^x ) by shifting up by 1.
Logarithmic Functions
Definition: ( f(x) = \log_a(x) ) where ( a > 0 ) and ( x > 0 ).
Relationship with exponentials: ( y = a^x ) can be rewritten as ( \log_a y = x ).
Properties
( \log_a(1) = 0 )
( \log_a(a) = 1 )
Solving Logarithmic Expressions
Convert between exponential and logarithmic forms.
Solve examples using base conversions and properties of logs.
Inverse Functions
The log function is the inverse of the exponential function.
Reflecting exponential graphs over ( y = x ) gives inverse graphs.
Graphs of Logarithmic Functions
Example: ( \log_6(x) ). Passes through ( (1,0) ) and ( (6,1) ).
Graph transformations apply similarly as exponential transformations.
Solving Logarithmic Equations
Use properties and rules to solve equations involving logs.
Rules of Logarithms
( \log_a(bc) = \log_a(b) + \log_a(c) )
( \log_a(\frac{b}{c}) = \log_a(b) - \log_a(c) )
( \log_a(b^n) = n \log_a(b) )
Examples
Calculate numerical expressions using log properties.
Solve equations involving both exponential and logarithmic forms.
Exponential Growth and Decay
Real-world applications: population growth, radioactive decay, etc.
Equations to model these phenomena: ( G = G_0 e^{kt} )
Solving Growth and Decay Problems
Determine initial values and find half-lives using exponential equations.
Experimental Data
Use logs to linearize exponential data for easier interpretation.
Logarithmic Models
Transforming exponential data to straight-line graphs through logarithmic scaling.
Example method: ( y = ax^b ) becomes ( \log(y) = b\log(x) + \log(a) ).
Applications
Utilize transformations and logs to solve and interpret complex equations in modeling.
Conclusion
Comprehensive coverage of logic and exponential functions.
Includes graph transformations, solving equations, and applications.
Note: The above notes summarize a lecture on exponential and logarithmic functions, including their properties, transformations, and applications in real-world scenarios.