MHF 4U Test 3 Comprehensive Guide: Logarithms, Exponentials, and Applications

Properties and Transformations of Logarithmic Functions

• Domain and Range Characteristics:

  • The domain of a basic logarithmic function $f(x) = \log_b(x)$ is restricted to $\{x \in \mathbb{R} \mid x > 0\}$.
  • For a transformed logarithmic function, the domain is not always all real numbers; it depends on the horizontal translations and reflections applied to the argument.

• Asymptotic Behavior:

  • Logarithmic functions possess a vertical asymptote. Unlike exponential functions, they do not have horizontal asymptotes.
  • The position of the vertical asymptote is directly affected by horizontal translations. If the function is shifted locally, the asymptote moves from $x = 0$ to the value of the horizontal shift.

• Order of Transformations:

  • Standard practice in function transformation dictates that horizontal and vertical stretches/compressions (and reflections) should be performed before horizontal and vertical translations to accurately map the function.

Solving Exponential and Logarithmic Equations

• Common Solution Strategies:

  • Exponential Form Conversion: Express the logarithmic equation in exponential form (e.g., $\log_b(x) = y \implies b^y = x$) and solve for the variable.
  • Logarithmic Laws Application: Use product, quotient, and power laws to simplify multi-term expressions into a single logarithm.
  • Quadratic substitution: Equations such as $3^{2x} = 7(3^x) - 12$ can be solved by treating $(3^x)$ as a single variable (e.g., let $u = 3^x$), resulting in a quadratic equation $u^2 - 7u + 12 = 0$.
  • Verification: It is essential to check for extraneous roots, mainly because the argument of a logarithm must be strictly positive.
  • Note: Squaring logarithmic expressions is not a standard or valid strategy for solving most logarithmic equations and typically introduces unnecessary complexity or incorrect roots.

• Specific Equation Types and Solutions:

  • Base Search: Solving $\log_x(81) = 4$. This converts to $x^4 = 81$. Since $3^4 = 81$, the solution is $x = 3$.
  • Exponential Subtraction: Solving $2^{x+2} - 2^x = 96$.
    • Factor out the common term: $2^x(2^2 - 1) = 96$.
    • Simplify: $2^x(4 - 1) = 96 \implies 2^x(3) = 96$.
    • Isolate the base: $2^x = 32$.
    • Solve for $x$: Since $2^5 = 32$, then $x = 5$.
  • Sum of Logs: Solving $\log(x + 3) + \log(x) = 1$.
    • Use Product Law: $\log(x(x + 3)) = 1$.
    • Convert to exponential form (base 10): $x^2 + 3x = 10^1$.
    • Rearrange: $x^2 + 3x - 10 = 0$.
    • Factor: $(x + 5)(x - 2) = 0$, yielding $x = -5, 2$.
    • Check constraints: $x = -5$ is extraneous because $\log(-5)$ is undefined. The final solution is $x = 2$.

Logarithmic Laws and Simplification Procedures

• Fundamental Laws:

  • Power Law: $\log_b(m^n) = n \log_b(m)$.
  • Product Law: $\log_b(m) + \log_b(n) = \log_b(mn)$.
  • Quotient Law: $\log_b(m) - \log_b(n) = \log_b(\frac{m}{n})$.

• Constraints on Laws:

  • Laws cannot be used to expand expressions like $\log_4(3y - 4)$ because the terms inside the argument are subtracted, not multiplied or divided.
  • These laws are only applicable when the base of each logarithmic term is identical.

• Calculations without a Calculator:

  • Evaluating $\log_2(16^3)$: $3 \log_2(16) = 3(4) = 12$.
  • Evaluating $\log(100^{-5})$: $-5 \log(100) = -5(2) = -10$.
  • Evaluating $\log(0.001^{-1})$: $-1 \log(10^{-3}) = -1(-3) = 3$.

• Single Logarithm Examples:

  • Simplifying $2\log(3) - \log(7) - \log(11)$:
    1. Power Law: $\log(3^2) - \log(7) - \log(11) = \log(9) - \log(7) - \log(11)$.
    2. Subtraction (Quotient Law): $\log(9) - (\log(7) + \log(11)) = \log(9) - \log(77)$.
    3. Final Result: $\log(\frac{9}{77})$.

Modeling Physical Phenomena

• Wind Speed in Tornadoes:

  • Formula: $S(d) = 300\log(d) + 65$, where $S(d)$ is speed in $mph$ and $d$ is distance in miles.
  • Scenario: Winds reach $400\,mph$. Estimate distance.
    • $400 = 300\log(d) + 65$
    • $335 = 300\log(d)$
    • $\log(d) = 1.1167$
    • $d = 10^{1.1167} \approx 13\,miles$.

• Radioactive Decay:

  • Given: Half-life $h = 7\,h$, Initial mass $2000\,g$.
  • Goal: Find the instantaneous rate of change in mass after $1.5\,days$ ($36\,h$).
  • Formula for mass: $M(t) = 2000(0.5)^{\frac{t}{7}}$.
  • The instantaneous rate of change is the derivative $M'(t)$, which involves the natural logarithm of the decay factor.

• Altimetry (Air Pressure):

  • Formula: $h = 18400 \log(\frac{P_0}{P})$.
  • $h$: Height above ground ($metres$).
  • $P$: Air pressure at height ($kPa$).
  • $P_0$: Air pressure on the ground at takeoff ($kPa$).
  • Application 1: Calculate height if $P = 32.5\,kPa$ and $P_0 = 102\,kPa$.
    • $h = 18400 \log(\frac{102}{32.5}) \approx 9132\,metres$.
  • Application 2: Calculate outside air pressure for a plane at $11\,000\,metres$ with ground pressure $102.5\,kPa$.
    • $11000 = 18400 \log(\frac{102.5}{P})$
    • $0.5978 = \log(\frac{102.5}{P})$
    • $10^{0.5978} = \frac{102.5}{P}$
    • $P \approx 25.9\,kPa$.
  • Application 3: Calculate height where air pressure is half of ground pressure ($P = 0.5P_0$).
    • $h = 18400 \log(\frac{P_0}{0.5P_0}) = 18400 \log(2) \approx 5539\,metres$.

Financial Modeling (Compound Interest)

• Timeline and Values:

  • Principal ($P$): $\$3000$
  • Value after 3 years ($A$): $\$4125.35$
  • Objective: Time required to triple the initial investment ($\$9000$).

• Calculation Steps:

  1. Find the Growth Rate ($i$):
     • $A = P(1+i)^n$
     • $4125.35 = 3000(1+i)^3$
     • $1.3751 = (1+i)^3 \implies 1+i = \sqrt[3]{1.3751} \approx 1.112$
  2. Solve for Tripled Time ($t$):
     • $9000 = 3000(1.112)^t$
     • $3 = (1.112)^t$
     • $\log(3) = t\log(1.112)$
     • $t = \frac{\log(3)}{\log(1.112)} \approx 10.3\,years$.

Graphical Relationships and Inverses

• Inverse Functions:

  • The inverse of an exponential function $f(x) = b^x$ is the logarithmic function $f^{-1}(x) = \log_b(x)$.
  • Graphs of inverses are reflections of the original function across the line $y = x$.

• Matching Procedure:

  • In the provided assessment, students must match exponential graphs with their corresponding logarithmic inverses based on their base values and growth/decay characteristics.
  • For each graph, identifying the equation of the original function allows for determining the inverse by swapping $x$ and $y$ coordinates and isolating the new $y$.