Logarithmic Functions, Exponential Growth/Decay, and Trigonometric Angle Fundamentals

Logarithmic and Exponential Functions

• Inverse Relationship of Logarithm and Exponential Functions:

  • Definition: $y = \log_b(x)$ if and only if $x = b^y$.

  • Function definition: $f(x) = \log_b(x)$ is the logarithmic function with base $b$.

  • Logarithmic and exponential functions with the same base $b$ are inverse functions of each other.

• Basic Properties of Logarithms:

  • $\log_a(a) = 1$: Asking what power $a$ must be raised to in order to get $a$. The answer is the $1\text{st}$ power.

  • $\log_a(1) = 0$: Asking what power $a$ must be raised to in order to get $1$. The answer is the $0\text{th}$ power.

  • $\log_a(a^x) = x$: Asking what power $a$ must be raised to in order to get $a^x$. The answer is $x$.

  • Inverse Function Property Viewpoint: Plugging an inverse function into the original function yields $x$.

  • Domain: Applies to all real numbers $x \in (-\infty, \infty)$ because the domain of the inner exponential function $a^x$ is all real numbers.

  • $a^{\log_a(x)} = x$: Inverse function property evaluated in the reverse order (inserting the logarithmic function inside the exponential function).

  • Domain: Restricted to $(0, \infty)$ because the domain of the inner logarithmic function $\log_a(x)$ is $(0, \infty)$.

• Domain and Range Derivation via Inverse Properties:

  • Exponential Function $y = a^x$ (for $a > 1$):

  • Domain: $(-\infty, \infty)$.

  • Range: $(0, \infty)$ (the graph stays strictly above the x-axis and never touches or crosses it).

  • Logarithmic Function $y = \log_a(x)$:

  • An inverse function swaps all $x$ and $y$ values.

  • Domain of exponential becomes Range of logarithm: $(-\infty, \infty)$.

  • Range of exponential becomes Domain of logarithm: $(0, \infty)$.

  • Undefined Values: Logarithms of $0$ and negative numbers are undefined in real numbers.

• Application Example: Doubling Money with Continuous Compounding

  • Problem Statement: Calculate how long it takes to double an investment earning $8\%$ compounded continuously.

  • Formula: Continuous compounding interest formula $A = P e^{r t}$.

  • Rate: $r = 0.08$

  • Target Condition: Find time $t$ when the future amount $A$ equals double the principal $P$ ($A = 2P$).

  • Step-by-step Solution:

  • Set up equation: $2P = P e^{0.08 t}$

  • Divide both sides by $P$: $2 = e^{0.08 t}$

  • Apply natural logarithm ($\ln$) to both sides: $\ln(2) = \ln(e^{0.08 t})$

  • Simplify using inverse property ($\ln$ and $e$ cancel): $\ln(2) = 0.08 t$

  • Solve for $t$: $t = \frac{\ln(2)}{0.08}$

  • Calculation: $t \approx 8.7\,\text{years}$ (rounded to one decimal place).

Logarithmic Rules and Properties

• General Assumptions: Let $b$, $M$, and $N$ be positive real numbers with $b \neq 1$, and let $r$ be any real number.

• Fundamental Rules of Logarithms:

  • Product Rule: $\log_b(M \times N) = \log_b(M) + \log_b(N)$

  • Reversible property: Can be used to expand a single logarithm into multiple terms or condense multiple logarithms into a single term.

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

  • Always structured as the log of the top minus the log of the bottom.

  • All logarithms must maintain identical bases on both sides of the equation.

  • Power Rule: $\log_b(M^r) = r \times \log_b(M)$

  • Proof for whole integer exponents using the product rule:

    • $\log_b(M^2) = \log_b(M \times M) = \log_b(M) + \log_b(M) = 2 \times \log_b(M)$

  • Generalization: Applies to any real number exponent $r$ (including negative, fractional, and irrational exponents), provided the exponent applies to the entirety of the inside argument.

• Change of Base Formula:

  • Let $a$, $b$, and $x$ be positive real numbers with $a \neq 1$ and $b \neq 1$.

  • Common Logarithm Form (Base 10): $\log_b(x) = \frac{\log(x)}{\log(b)}$

  • Natural Logarithm Form (Base $e$): $\log_b(x) = \frac{\ln(x)}{\ln(b)}$

  • Arbitrary Base Form (Base $a$): $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$

  • Terminology and Conventions:

  • If no base is explicitly written on a logarithm (e.g., $\log(x)$), it is understood to be a common log with base 10 (similar to how square root radical indices are omitted).

  • Natural logarithms are denoted as $\ln(x)$ (base $e$).

  • Natural logs and common logs are primarily used to evaluate non-standard base logarithms on scientific calculators.

  • Conversion to an arbitrary base $a$ is primarily used when solving algebraic equations that contain mixed logarithmic bases.

• Change of Base Evaluation Example: Evaluate $\log_7(12)$ rounded to 4 decimal places.

  • Using Common Logarithms:

  • $\log_7(12) = \frac{\log(12)}{\log(7)} \approx 1.2770$

  • Using Natural Logarithms:

  • $\log_7(12) = \frac{\ln(12)}{\ln(7)} \approx 1.2770$

  • Conclusion: Both common logs and natural logs yield the identical numerical output. Natural log notation is frequently preferred due to writing brevity.

Exponential Growth and Decay

• Universal Exponential Growth and Decay Model:

  • Formula: $A(t) = A_0 e^{k t}$

  • Variable Definitions:

  • $A(t)$: Amount of substance, population, or quantity remaining at time $t$.

  • $A_0$: Initial amount at time $t = 0$ (i.e., $A(0)$).

  • $t$: Time elapsed.

  • $k$: Constant rate of growth or decay.

  • Behavior of Rate Constant $k$:

  • If $k > 0$, the model represents exponential growth.

  • If $k < 0$, the model represents exponential decay.

  • Using a single generalized equation avoids needing separate formulas for growth versus decay.

• Application Example: Half-Life Determination of a Radioactive Element

  • Problem Statement: An experiment begins with $36\,\text{g}$ of a radioactive element, which decays to $9\,\text{g}$ in $24\,\text{hours}$. Calculate its half-life to the nearest hour.

  • Core Concept: Half-life is the time $t$ required for the remaining amount $A(t)$ to equal half of the initial amount $A_0$ ($A(t) = \frac{1}{2} A_0$).

  • Initial Given Information:

  • Initial amount $A_0 = 36\,\text{g}$.

  • Half-life target amount $A(t) = 18\,\text{g}$.

  • Decay data point: $A(24) = 9\,\text{g}$ at $t = 24\,\text{hours}$.

  • Step 1: Solve for the decay rate constant $k$.

  • Set up equation with decay data point: $9 = 36 e^{k \times 24}$

  • Divide both sides by $36$: $0.25 = e^{24 k}$

  • Apply natural logarithm to both sides: $\ln(0.25) = \ln(e^{24 k})$

  • Simplify using inverse properties: $\ln(0.25) = 24 k$

  • Express exact rate constant $k$: $k = \frac{\ln(0.25)}{24}$

  • Note: Keep $k$ in exact fraction form to prevent premature rounding error in multi-step problems.

  • Step 2: Solve for the half-life time $t$ when $A(t) = 18\,\text{g}$.

  • Substitute $A(t) = 18$ and exact $k$ into model: $18 = 36 e^{\left(\frac{\ln(0.25)}{24}\right) t}$

  • Divide both sides by $36$: $0.5 = e^{\left(\frac{\ln(0.25)}{24}\right) t}$

  • Apply natural logarithm to both sides: $\ln(0.5) = \frac{\ln(0.25)}{24} t$

  • Multiply by reciprocal to isolate $t$: $t = \frac{24 \times \ln(0.5)}{\ln(0.25)}$

  • Exact Solution: $t = \frac{24 \times \ln(0.5)}{\ln(0.25)}$

  • Evaluated Numerical Value: $t = 12\,\text{hours}$ (rounded to the nearest hour).

  • Verification of Results:

  • At $t = 0\,\text{hours}$, amount is $36\,\text{g}$.

  • After $12\,\text{hours}$ ($1$ half-life), amount reduces by half to $18\,\text{g}$.

  • After $24\,\text{hours}$ ($2$ half-lives), amount reduces by half again to $9\,\text{g}$. This precisely matches the problem statement.

Angle Basics and Standard Position

• Course Scope Note: Exponential and logarithmic review material is situated in tutorial modules. Section 5.1 marks the start of Test 1 core material.

• Fundamental Geometric Definitions:

  • Ray: A portion of a line bounded by a single endpoint and extending infinitely in one direction.

  • Angle: Formed by rotating a ray about its endpoint.

  • Initial Side: The original starting position of the ray prior to rotation.

  • Terminal Side: The position of the ray following rotation.

  • Vertex: The fixed endpoint about which the ray rotates.

• Rotational Direction and Sign Conventions:

  • Counterclockwise Rotation: Results in a positive angle measure.

  • Clockwise Rotation: Results in a negative angle measure.

• Coterminal Angles:

  • Definition: Angles that share the exact same initial side and exact same terminal side.

  • Example: Angle $\alpha$ rotates counterclockwise from the initial side to the terminal side (positive angle). Angle $\beta$ rotates clockwise around the circle to end at the exact same terminal side (negative angle). Thus, $\alpha$ and $\beta$ are coterminal angles.

• Standard Position in Rectangular Coordinates:

  • Definition: An angle is in standard position when its vertex is located at the origin $(0,0)$ and its initial side lies along the positive x-axis.

  • Quadrantal Location: An angle in standard position lies in a specific quadrant if its terminal side resides within that quadrant.

  • Quadrant Layout (Counterclockwise ordering):

  • Quadrant I: Upper right ($x > 0$, $y > 0$).

  • Quadrant II: Upper left ($x < 0$, $y > 0$).

  • Quadrant III: Lower left ($x < 0$, $y < 0$).

  • Quadrant IV: Lower right ($x > 0$, $y < 0$).

  • Quadrantal Angles:

  • Definition: An angle in standard position whose terminal side falls directly on a coordinate axis (x-axis or y-axis).

  • Quadrantal angles lie on the boundary lines between quadrants and do not belong to any single quadrant.

• Degree Angle Classification:

  • $1^\circ$ (One Degree): Defined as a rotation of $\frac{1}{360}$ of a complete counterclockwise revolution about the vertex.

  • One Complete Revolution = $360^\circ$

  • Acute Angle: Angle measure strictly between $0^\circ$ and $90^\circ$ (e.g., $40^\circ$).

  • Right Angle: Angle measure equal to exactly $90^\circ$ ($\frac{1}{4}$ complete revolution). Indicated graphically by a small square at the vertex.

  • Obtuse Angle: Angle measure strictly between $90^\circ$ and $180^\circ$ (e.g., $135^\circ$).

  • Straight Angle: Angle measure equal to exactly $180^\circ$ ($\frac{1}{2}$ complete revolution, forming a straight line).

Degrees, Minutes, Seconds (DMS) Notation

• Overview: Standard system used in navigation and positional measurement. Analogous to time measurements where degrees correspond to hours.

• Fundamental Conversion Equivalencies:

  • $1^\circ = 60'$ (1 degree equals 60 minutes; denoted by single apostrophe $'$).

  • $1' = 60''$ (1 minute equals 60 seconds; denoted by double quotation mark $''$).

• Conversion Process: DMS to Decimal Degrees

  • Example: Convert $50^\circ 45' 36''$ into decimal degree notation rounded to two decimal places.

  • Step 1: Separate into component terms: $50^\circ + 45' + 36''$.

  • Step 2: Convert seconds to minutes via unit factor $\left(\frac{1'}{60''}\right)$:

  • $36'' \times \frac{1'}{60''} = 0.6'$

  • Step 3: Add converted minutes to existing minutes:

  • $45' + 0.6' = 45.6'$

  • Step 4: Convert total minutes to degrees via unit factor $\left(\frac{1^\circ}{60'}\right)$:

  • $45.6' \times \frac{1^\circ}{60'} = 0.76^\circ$

  • Step 5: Add converted degrees to integer degrees:

  • $50^\circ + 0.76^\circ = 50.76^\circ$

• Conversion Process: Decimal Degrees to DMS

  • Example: Convert $120.64^\circ$ to degrees, minutes, seconds notation.

  • Step 1: Separate integer degrees and decimal degrees:

  • $120^\circ + 0.64^\circ$

  • Step 2: Convert decimal degrees to minutes via unit factor $\left(\frac{60'}{1^\circ}\right)$:

  • $0.64^\circ \times \frac{60'}{1^\circ} = 38.4'$

  • Step 3: Separate integer minutes and decimal minutes:

  • $38' + 0.4'$

  • Step 4: Convert decimal minutes to seconds via unit factor $\left(\frac{60''}{1'}\right)$:

  • $0.4' \times \frac{60''}{1'} = 24''$

  • Step 5: Combine into standard DMS string format:

  • $120^\circ 38' 24''$

Radian Measure and Conversions

• Definitions:

  • Central Angle: An angle whose vertex is positioned at the center of a circle.

  • Radian Measure ($1\,\text{radian}$): The measure of a positive central angle that intercepts an arc length $s$ equal to the radius $r$ of the circle.

  • Arc Length Ratio Formula:

  • $\theta = \frac{s}{r}$

  • Where $\theta$ is the central angle measure in radians, $s$ is arc length, and $r$ is radius.

• Dimensionless Nature of Radians:

  • Radians are a ratio of two lengths ($\frac{\text{length}}{\text{length}}$), rendering units algebraically dimensionless.

  • Unit Convention: Degrees MUST always feature the degree symbol ($^\circ$). Any angle lacking a specified unit symbol is mathematically treated as being in radians.

• Revolution Equivalencies and Conversion Factors:

  • Circumference of a circle: $C = 2 \pi r$.

  • One complete revolution in radians = $2\pi\,\text{radians}$.

  • Equivalence relation: $360^\circ = 2\pi\,\text{radians} \implies 180^\circ = \pi\,\text{radians}$.

  • Degrees to Radians Conversion Factor: Multiply by $\frac{\pi\,\text{radians}}{180^\circ}$.

  • Radians to Degrees Conversion Factor: Multiply by $\frac{180^\circ}{\pi\,\text{radians}}$.

• Degree-to-Radian Conversion Examples:

  • Convert $30^\circ$: $30^\circ \times \frac{\pi}{180^\circ} = \frac{30\pi}{180} = \frac{\pi}{6}$

  • Convert $180^\circ$: $180^\circ \times \frac{\pi}{180^\circ} = \pi$

  • Convert $-45^\circ$: $-45^\circ \times \frac{\pi}{180^\circ} = -\frac{45\pi}{180} = -\frac{\pi}{4}$

  • Convert $360^\circ$: $360^\circ \times \frac{\pi}{180^\circ} = 2\pi$

• Radian-to-Degree Conversion Examples:

  • Convert $\frac{2\pi}{3}$: $\frac{2\pi}{3} \times \frac{180^\circ}{\pi} = \frac{360^\circ}{3} = 120^\circ$

  • Convert $-\frac{5\pi}{4}$: $-\frac{5\pi}{4} \times \frac{180^\circ}{\pi} = -\frac{900^\circ}{4} = -225^\circ$

  • Convert $3\,\text{radians}$: $3 \times \frac{180^\circ}{\pi} = \frac{540^\circ}{\pi}$ (Exact angle solution; degree symbol mandatory).

Complementary and Supplementary Angles

• Basic Definitions:

  • Complementary Angles: Two positive angles whose sum equals $90^\circ$ (or $\frac{\pi}{2}$ radians).

  • Supplementary Angles: Two positive angles whose sum equals $180^\circ$ (or $\pi$ radians).

• Complement and Supplement Evaluation Examples:

  • Example 1: Degree Given Angle ($40^\circ$)

  • Complement: $90^\circ - 40^\circ = 50^\circ$

  • Supplement: $180^\circ - 40^\circ = 140^\circ$

  • Example 2: Radian Given Angle ($\frac{\pi}{3}$)

  • Formatting Rule: Answers must remain in the original input unit (radians).

  • Complement Calculation:

    • Equivalent threshold: $90^\circ = \frac{\pi}{2}$

    • Formula: $\frac{\pi}{2} - \frac{\pi}{3}$

    • Determine common denominator ($6$): $\frac{3\pi}{6} - \frac{2\pi}{6} = \frac{\pi}{6}$

  • Supplement Calculation:

    • Equivalent threshold: $180^\circ = \pi$

    • Formula: $\pi - \frac{\pi}{3} = \frac{\pi}{1} - \frac{\pi}{3}$

    • Determine common denominator ($3$): $\frac{3\pi}{3} - \frac{\pi}{3} = \frac{2\pi}{3}$

• Practical Practice Note: Proficiency in fraction arithmetic (common denominators, simplifying rational expressions) is crucial for subsequent trigonometry chapters.