Study Notes on Exponential Functions and Logarithms

To the zero power is defined as one: If $a^0 = 1$ for any non-zero number a, this is fundamental when understanding exponential functions.
Example Given: Evaluate an exponential function, leading to a value such as 644390% when plugging into a calculator after determining a base value.
Contextualizing calculations: Reference to understanding values at specific instances, such as eight weeks later.

Basic task: Evaluate the function at provided values or solve the equation.
Solving for x when y is a certain value, such as halving the original value.
Example: If the original sales were 10,000, then half would be 5,000.
Formulation of the equation: $y = 10,000 imes 3^{-0.5x}$ captures the decay relationship.

Review different types of functions: Exponential functions vs. Power functions vs. Quadratics.
Link: Exponential decay is characterized by a variable in the exponent.
Solving for x involves isolating the exponent through algebraic manipulation, logarithms, and possibly software tools like Desmos to visualize solutions.
Graphing technique: Overlay two equations, $y = 10,000 imes 3^{-0.5x}$ and $y = 5,000$ to find intersections visually and determine solutions:
- The exponential decay graph approaches zero, starting from the initial 10,000 and decreasing.
- Significant intersection point found at $x ext{ approximately } 12.6186$ weeks, where sales reach 5,000.

Reflection on exponential bases: A base greater than one with a negative exponent results in decay.
Example: Calculate the value transforms based on understanding the exponential and its decay direction.
For a base of 3 raised to a negative exponent ( ext{-0.5}), effectively results in decay behaviour, similarly solvable via logarithmic transformations.

Example of Population Growth: Model given as $y = 53,000 e^{0.0152t}$ with an initial size of 52,000.
Analyzing whether the population is increasing/decreasing based on parameters:
- Base e is positive leading to growth.
- Approximate inquiry: Estimate population for 2005 (5 years after initial): Plugging into the model yields approximately 57,127 individuals.

Example: To find when population will reach 60,000, set up the equation corresponding to the model.
- Find graphically by setting the growth equation against a constant, leading to intersections signifying time.
Utilizing both visual methods and algebraic checks to validate calculations.

Introduction to Inverses: Logarithmic functions are the inverse of exponential functions.
- For any point $(a,b)$ on function $f$ , the inverse point becomes $(b,a)$ on $f^{-1}$ .
Applications of logarithms span various fields: Example contexts include measuring sound, earthquake levels, pH in solutions, etc.
Fundamental rule for logarithms: It only applies if the base is greater than zero and not equal to one., which maintains valid transformations.

Domain for exponential functions: All real numbers.
Range: Positively only, from zero to infinity, excluding the asymptote of y=0.
Logarithmic functions: Domain restricted to positive values (0, ∞) while the range spans (-∞, ∞).

For any base greater than one (e.g., $1.5^x$ or $2^x$ ), the shape of the graph will remain similar across all such bases. Likewise for logarithms, consistently displaying similar forms depending on the base.
Highlight about similarity: All exponential graphs through the point (0,1) unless scaled or transformed otherwise.

Different bases yield different steepness in their graphs but maintain similar core properties.
The understanding of inverse relationships simplifies into transitionary forms across function families:
- y = a^x
ightarrow y = ext{log}_a(x).
Key special cases: Natural log (ln) corresponding to base e, common log typically understood to have base 10.

Understanding the relationship between exponentials and logs is crucial for problem-solving.
The context of practical applications and mathematical properties ensures a firm grasp for examination preparation.