Precalculus Study Guide: Logarithms, Exponentials, Sequences, and Financial Applications

Logarithmic and Exponential Functions

Properties and Operations of Logarithms

Logarithms are the inverses of exponential functions and are essential for solving equations involving exponentials. Understanding their properties allows for simplification and manipulation of expressions.

• Product Rule: $\log_b(xy) = \log_b(x) + \log_b(y)$

• Quotient Rule: $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$

• Power Rule: $\log_b(x^k) = k\log_b(x)$

• Change of Base Formula: $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$

Example:

• Expand $\log_2(16x/40)$: $\log_2(16x) - \log_2(40) = \log_2(16) + \log_2(x) - \log_2(40)$

• Simplify $\log_2(x) + \log_2(y) = \log_2(xy)$

Solving Logarithmic and Exponential Equations

Equations involving logarithms and exponentials often require the use of their properties for simplification and solution.

• To solve $e^{2x} = 3$:

  • Take the natural logarithm of both sides: $2x = \ln(3)$

  • So, $x = \frac{\ln(3)}{2}$

• To solve $\log_2(x+3) = 2 - \log_2(x+6)$:

  • Rewrite: $\log_2(x+3) + \log_2(x+6) = 2$

  • Combine: $\log_2((x+3)(x+6)) = 2$

  • So, $(x+3)(x+6) = 2^2 = 4$

Domain and Range of Logarithmic and Exponential Functions

The domain and range of these functions are determined by their algebraic structure.

• Exponential Function $f(x) = e^{x} + A$:

  • Domain: $(-\infty, \infty)$

  • Range: $(A, \infty)$ if $A > 0$

• Logarithmic Function $f(x) = \ln(x^2 + 49)$:

  • Domain: $(-\infty, \infty)$ (since $x^2 + 49 > 0$ for all real $x$)

Sequences and Series

Arithmetic Sequences

An arithmetic sequence is a sequence of numbers with a constant difference between consecutive terms.

• General Term: $a_n = a_1 + (n-1)d$

• Sum of First $n$ Terms: $S_n = \frac{n}{2}(a_1 + a_n)$ or $S_n = \frac{n}{2}[2a_1 + (n-1)d]$

Example:

• Given $-10, -6, -2, 2, ...$, $d = 4$, $a_1 = -10$

• General term: $a_k = -10 + (k-1)4$

• Sum of first 8 terms: $S_8 = \frac{8}{2}[2(-10) + (8-1)4]$

Geometric Sequences

A geometric sequence is a sequence where each term is found by multiplying the previous term by a constant ratio.

• General Term: $a_k = a \cdot r^{k-1}$

• Sum of First $n$ Terms: $S_n = \frac{a(1 - r^n)}{1 - r}$

• Sum of Infinite Convergent Series: $S = \frac{a}{1 - r}$, for $|r| < 1$

Financial Mathematics: Compound Interest and Effective Rate

Compound Interest

Compound interest calculations are fundamental in financial mathematics, especially for savings, loans, and investments.

• Future Value (Periodic Compounding): $S = P(1 + r)^n$

• Present Value (Periodic Compounding): $P = \frac{S}{(1 + r)^n}$

• Future Value (Continuous Compounding): $S = Pe^{rt}$

• Present Value (Continuous Compounding): $P = Se^{-rt}$

Example:

• To find the time for an investment to triple at 15% continuous compounding: $3P = P e^{0.15t} \implies t = \frac{\ln(3)}{0.15}$

Effective Rate

The effective rate is the actual interest rate earned or paid after accounting for compounding.

• Effective Rate (Periodic Compounding): $r_e = (1 + \frac{r}{n})^n - 1$

• Effective Rate (Continuous Compounding): $r_e = e^r - 1$

Inverse Functions

Finding the Inverse of a Function

To find the inverse of a one-to-one function, solve for $x$ in terms of $y$ and interchange the variables.

• Example: For $f(x) = Ae^{Bx+C} - D$, solve $y = Ae^{Bx+C} - D$ for $x$:

• $y + D = Ae^{Bx+C}$

• $\frac{y + D}{A} = e^{Bx+C}$

• $\ln\left(\frac{y + D}{A}\right) = Bx + C$

• $x = \frac{1}{B}\left[\ln\left(\frac{y + D}{A}\right) - C\right]$

Domain and Range of Inverse Functions

The domain of the inverse function is the range of the original function, and vice versa.

• Domain of $f^{-1}$: Range of $f$

• Range of $f^{-1}$: Domain of $f$

Summary Table: Key Formulas

Concept	Formula	Description
Arithmetic Sequence (nth term)	$a_n = a_1 + (n-1)d$	General term
Arithmetic Sequence (sum)	$S_n = \frac{n}{2}(a_1 + a_n)$	Sum of first n terms
Geometric Sequence (nth term)	$a_k = a r^{k-1}$	General term
Geometric Sequence (sum)	$S_n = \frac{a(1 - r^n)}{1 - r}$	Sum of first n terms
Geometric Series (infinite sum)	$S = \frac{a}{1 - r}$	Convergent series, $|r| < 1$
Future Value (periodic)	$S = P(1 + r)^n$	Compound interest
Present Value (periodic)	$P = \frac{S}{(1 + r)^n}$	Compound interest
Future Value (continuous)	$S = Pe^{rt}$	Continuous compounding
Present Value (continuous)	$P = Se^{-rt}$	Continuous compounding
Effective Rate (periodic)	$r_e = (1 + \frac{r}{n})^n - 1$	n times per year
Effective Rate (continuous)	$r_e = e^r - 1$	Continuous compounding

Additional info:

• Some context and explanations have been expanded for clarity and completeness.

• Examples and step-by-step solutions are provided to illustrate key concepts.