BackCollege Algebra Chapter 2 Study Notes: Exponents, Logarithms, and Exponential Growth
Study Guide - Smart Notes
Tailored notes based on your materials, expanded with key definitions, examples, and context.
Properties of Exponents
Definition and Key Concepts
An exponent (also called power or degree) tells us how many times the base will be multiplied by itself. For example, in , the exponent is 5 and the base is x. This means that x will be multiplied by itself 5 times: .
Product Rule: Same base, add exponents:
Quotient Rule: Same base, subtract exponents:
Power Rule: Power raised to a power, multiply exponents:
Power Rule II: Product to a power, distribute the power to each base:
Negative Exponent I: Flip and change the sign to positive:
Negative Exponent II: Flip and change sign to positive:
Zero Exponent: Anything to the zero power (except 0) is one:
Warning: cannot be simplified to unless the bases are the same.
Common Errors Involving Exponents
(be careful with parentheses)
Examples
Rational Exponents and Roots
Definition and Properties
If is a real number greater than zero and is a positive integer, then:
Fun Fact: , but .
Examples
Writing with Positive Exponents
Logarithms
Definition and Properties
For , , and , if and only if . The function is called the logarithmic function with base .
for any real number
for any
Converting Between Exponential and Logarithmic Form
Exponential to Logarithmic:
Logarithmic to Exponential:
Examples
Common and Natural Logarithms
The logarithm with base 10 is called the common logarithm and is denoted by .
The logarithm with base is called the natural logarithm and is denoted by .
Base 10 | Base e |
|---|---|
log(10) = 1 | ln(e) = 1 |
log(1) = 0 | ln(1) = 0 |
log(10^x) = x | ln(e^x) = x |
10^{log(x)} = x | e^{ln(x)} = x |
Examples (Without Calculator)
Change of Base Formula
To evaluate logarithms with bases other than 10 or , use the change of base formula:
Example:
Solving Exponential and Logarithmic Equations
General Steps
Isolate the exponential or logarithmic expression on one side.
Convert between exponential and logarithmic form as needed.
Solve for the variable.
Examples
Applications: Exponential and Logarithmic Models
Recall and Decay Problems
Exponential and logarithmic functions are used to model real-world phenomena such as memory recall, population growth, and radioactive decay.
Example: If represents the number of days until students recall important features of a lecture:
To find when 90% recall: , solve for .
To find when 50% recall: , solve for .
Regression and Data Modeling
Logarithmic regression can be used to model data, such as the relationship between age and hours of sleep required.
Age (years) | Hours of Sleep |
|---|---|
3 | 14 |
8 | 10 |
14 | 9 |
25 | 8 |
45 | 7 |
Regression equation:
To find hours of sleep for a 35-year-old: hours
To find age for 12 hours of sleep: years
Properties of Logarithms
Examples: Condensing Logarithms
Exponential Growth
Definition and Formula
Exponential growth describes processes that increase by a constant percentage or factor over equal time intervals.
General formula:
= amount at time
= initial amount (when )
= growth factor ()
Examples
Population triples every time period:
Money grows by 5% per year:
Population increases 3.4% per year:
Summary Table: Exponent and Logarithm Properties
Property | Exponent Form | Logarithm Form |
|---|---|---|
Product | ||
Quotient | ||
Power | ||
Zero | ||
Identity |
Additional info: These notes cover foundational concepts in exponents, roots, logarithms, and exponential growth, which are essential for success in College Algebra. The examples and tables provide practical applications and reinforce the properties discussed.
