College Algebra Chapter 2 Study Notes: Exponents, Logarithms, and Exponential Growth

Properties of Exponents

Definition and Key Concepts

An exponent (also called power or degree) indicates how many times a base will be multiplied by itself. For example, in $x^5$, the exponent is 5 and the base is x, meaning $x$ is multiplied by itself 5 times.

• Product Rule: Same base, add exponents: $a^m \cdot a^n = a^{m+n}$

• Quotient Rule: Same base, subtract exponents: $\frac{a^m}{a^n} = a^{m-n}$

• Power Rule: Power raised to a power, multiply exponents: $(a^m)^n = a^{mn}$

• Power to Product Rule: Distribute exponent to each factor: $(ab)^n = a^n b^n$

• Power to Quotient Rule: Distribute exponent to numerator and denominator: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

• Negative Exponent: Flip and change the sign: $a^{-n} = \frac{1}{a^n}$

• Zero Exponent: Any nonzero base to the zero power is 1: $a^0 = 1$ (for $a \neq 0$)

Property	General Form	Application	Example
Product Rule	$a^m \cdot a^n$	$a^{m+n}$	$x^2 \cdot x^3 = x^5$
Quotient Rule	$\frac{a^m}{a^n}$	$a^{m-n}$	$x^5 / x^2 = x^3$
Power Rule	$(a^m)^n$	$a^{mn}$	$(x^2)^3 = x^6$
Power to Product	$(ab)^n$	$a^n b^n$	$(2x)^3 = 8x^3$
Power to Quotient	$\left(\frac{a}{b}\right)^n$	$\frac{a^n}{b^n}$	$\left(\frac{x}{y}\right)^2 = \frac{x^2}{y^2}$
Negative Exponent	$a^{-n}$	$\frac{1}{a^n}$	$x^{-3} = \frac{1}{x^3}$
Zero Exponent	$a^0$	$1$	$(4x^2)^0 = 1$

Warning: $\frac{a^m}{b^m}$ cannot be simplified unless the bases are the same.

Common Errors Involving Exponents

• $a^n \neq (a^n)$ (do not confuse with parentheses)

• $(ab)^n \neq (a^n)(b^n)$ if $a$ and $b$ are not both positive

• $a^n + a^m \neq a^{n+m}$

Examples: Simplifying Exponential Expressions

• Example: $(6x^7n^5)(8x^4n^6) = 48x^{11}n^{11}$

• Example: $\frac{25a^{10}}{b^2}$

• Example: $\frac{d^8}{q^4z^4}$

• Example: $\frac{-6b^7}{12a^7} = -\frac{b^7}{2a^7}$

• Example: $\frac{p^{10}p^{12}}{q^4} = \frac{p^{22}}{q^4}$

Rational Exponents and Roots

Definition and Properties

For $a > 0$, $n > 0$, and $r$ rational, $a^{1/n} = \sqrt[n]{a}$ and $a^{m/n} = (\sqrt[n]{a})^m$.

• Example: $100^{1/2} = \sqrt{100} = 10$

• Example: $x^{3/4} = (x^{1/4})^3$

• Example: $\sqrt{x} = x^{1/2}$

Logarithms

Definition and Properties

A logarithm is the inverse operation to exponentiation. For $x > 0$, $a > 0$, $a \neq 1$, $y = \log_a x$ if and only if $a^y = x$.

• Product Rule: $\log_a(MN) = \log_a M + \log_a N$

• Quotient Rule: $\log_a\left(\frac{M}{N}\right) = \log_a M - \log_a N$

• Power Rule: $\log_a(M^p) = p \log_a M$

• Change of Base Formula: $\log_b a = \frac{\log_c a}{\log_c b}$

Special Logarithms

• Common Logarithm: Base 10, $\log_{10} x = \log x$

• Natural Logarithm: Base $e$, $\log_e x = \ln x$

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: Converting Between Exponential and Logarithmic Form

• Exponential to Logarithmic: $3^2 = 9 \rightarrow \log_3 9 = 2$

• Logarithmic to Exponential: $\log_2(x) = 4 \rightarrow 2^4 = x$

Examples: Evaluating Logarithms Without a Calculator

• $\log_{10}(10^{15}) = 15$

• $10^{\log_{10}(2.14)} = 2.14$

• $\ln(e^6) = 6$

Change of Base Formula

To evaluate logarithms with bases other than 10 or $e$:

• $\log_b a = \frac{\ln a}{\ln b}$ or $\frac{\log a}{\log b}$

• Example: $\log_6 38 = \frac{\ln 38}{\ln 6} \approx 2.03018$

Solving Exponential and Logarithmic Equations

General Steps

• Isolate the exponential or logarithmic expression.

• Convert to the other form (exponential to logarithmic or vice versa).

• Solve for the variable.

Examples

• $10 + 5^x = 100 \rightarrow 5^x = 90 \rightarrow x = \log_5 90 \approx 2.80$

• $3x = \log_7 21 \rightarrow x = \frac{\log_7 21}{3} \approx 0.52$

• $2 - \log(x) = 5 \rightarrow \log(x) = -3 \rightarrow x = 10^{-3} = 0.001$

Applications: Modeling with Exponential and Logarithmic Functions

Recall and Memory Model

Example: $P(t) = 95 - 30 \log_2(t)$, where $P(t)$ is the percent of information recalled after $t$ days.

• To find when 90% is recalled: $90 = 95 - 30 \log_2(x) \rightarrow x = 2^{1/6} \approx 1.1$ days

• To find when 50% is recalled: $50 = 95 - 30 \log_2(x) \rightarrow x = 2^{3/2} \approx 2.8$ days

Logarithmic Regression Example

Given a table of age vs. hours of sleep, the regression equation is $y = 17.3 - 3.3 \ln(x)$.

• For a 35-year-old: $y = 17.3 - 3.3 \ln(35) \approx 5.6$ hours

• For 12 hours of sleep: $12 = 17.3 - 3.3 \ln(x) \rightarrow x = e^{-5.3/3.3} \approx 5.0$ years

Properties of Logarithms: Condensing and Expanding

• $\log_a(M) + \log_a(N) = \log_a(MN)$

• $\log_a(M) - \log_a(N) = \log_a\left(\frac{M}{N}\right)$

• $p \log_a(M) = \log_a(M^p)$

Examples

• $\log(25) + \log(2) = \log(50)$

• $\ln(3^7) - \ln(3) = \ln(3^6)$

• $\log_5(x+4) + \log_5(3x+1) = \log_5((x+4)(3x+1))$

Exponential Growth

Definition and Formula

Exponential growth occurs when a quantity increases by a fixed percentage over equal time intervals. The general formula is:

$y = C a^x$

• $y$ = amount at time $x$

• $C$ = initial amount

• $a$ = growth factor ($a > 1$)

Examples

• Population of 5,600 triples every time period: $y = 5600 \cdot 3^x$

• Initial amount $45,000$ with 5% growth: $y = 45000 \cdot 1.05^x$

• Population of 53 million increases 3.4% per year: $y = 53 \cdot 1.034^x$

Summary Table: Key Properties and Formulas

Concept	Formula	Example
Product Rule (Exponents)	$a^m \cdot a^n = a^{m+n}$	$x^2 \cdot x^3 = x^5$
Quotient Rule (Exponents)	$\frac{a^m}{a^n} = a^{m-n}$	$x^5 / x^2 = x^3$
Power Rule (Exponents)	$(a^m)^n = a^{mn}$	$(x^2)^3 = x^6$
Product Rule (Logs)	$\log_a(MN) = \log_a M + \log_a N$	$\log_2(8 \cdot 4) = \log_2 8 + \log_2 4$
Quotient Rule (Logs)	$\log_a\left(\frac{M}{N}\right) = \log_a M - \log_a N$	$\log_3(9/3) = \log_3 9 - \log_3 3$
Power Rule (Logs)	$\log_a(M^p) = p \log_a M$	$\log_5(2^3) = 3 \log_5 2$
Change of Base	$\log_b a = \frac{\ln a}{\ln b}$	$\log_6 38 = \frac{\ln 38}{\ln 6}$
Exponential Growth	$y = C a^x$	$y = 1000 \cdot 1.05^x$

Additional info: These notes cover foundational concepts in exponents, logarithms, and exponential growth, which are essential for success in College Algebra. Mastery of these properties and their applications is critical for solving equations and modeling real-world phenomena.