Exponential and Logarithmic Functions Flashcards

Geometric Sequences and Series

• A geometric sequence is defined by the formula: ( a_n = a_1(r)^{n-1} ), where ( a_n ) is the nth term, ( a_1 ) is the first term, ( r ) is the common ratio, and ( n ) is the term number.
• A geometric series sums the terms of a geometric sequence and is calculated with ( S_n = \frac{a_1(1 - r^n)}{1 - r} ), where ( n ) is the number of terms.
• The infinite geometric series applies when ( r < 1 ) and is calculated with ( S = \frac{A_1}{1 - r} ).

Exponential Functions

• Exponential growth is modeled by the equation ( A = P(1 + r)^t ), where ( A ) is the amount after time ( t ), ( P ) is the initial amount, and ( r ) is the growth rate.
• Exponential decay shares a similar formula: ( A = P(1 - r)^t ), demonstrating how quantities decrease over time.
• Compound interest formula is represented by ( A = P(1 + \frac{r}{n})^{nt} ), incorporating the compounding frequency ( n ).

Rates and Growth Factors

• In a situation where bacteria double every half hour, the growth rate is 100%, and the growth factor is 200% for that half-hour period; hence the hourly growth rate becomes 300%, leading to an hourly growth factor of 400%.

Identifying Exponential Behavior

• To determine if a table shows exponential behavior, check if there is a constant ratio ( \frac{A_n}{A_{n-1}} ) between consecutive terms, indicating a multiplicative relationship rather than a constant slope.

Properties of Exponents

• The rules of exponents, including ( a^x \cdot b^x = (ab)^x ), demonstrate how to simplify expressions involving multiplication of powers.
• Exponentiation also adheres to ( (b^n)^m = b^{n \cdot m} ) and ( a^x \cdot a^y = a^{x+y} ).
• For multiplying different bases raised to the same power, the rule is ( (ab)^x = (a^x)(b^x) ).

Additional Insights

• The geometric series can be utilized even when ( r > 1 ) to find the sum of a finite number of terms but should be approached with caution for contexts involving infinite series.