Pre-Calculus Chapter 3 Review Notes

Chapter 3: Exponential & Logarithmic Functions

Lesson 3.1: Exponential Functions and Their Graphs

• Key Features:

  • Domain: All real numbers $(- ext{infinity}, ext{infinity})$

  • Range: Positive real numbers $(0, ext{infinity})$(function never reaches zero)

  • Asymptote: Horizontal asymptote at $y = 0$

  • Intercepts:

  • $y$-intercept at (0, 1)

  • Transformations:

  • Shifts, reflections, stretches, and shrinks must be noted for accurate graph representation.

Lesson 3.2: Logarithmic Functions and Their Graphs

• Key Features:

  • Domain: Positive real numbers $(0, ext{infinity})$

  • Range: All real numbers $(- ext{infinity}, ext{infinity})$

  • Asymptote: Vertical asymptote at $x = 0$

  • Intercepts:

  • $x$-intercept at (1, 0)

  • Transformations:

  • Similar transformations applied as in exponential functions.

Lesson 3.3: Properties of Logarithms

• Change of Base Formula:

  • $ext{If } b ≠ 1, ext{ then } rac{ ext{log}<em>a x}{ ext{log}</em>a b} = ext{log}_b x$

• Expanding and Condensing Logs:

  • Ability to expand and condense using properties such as:

  • $ext{log}<em>b MN = ext{log}</em>b M + ext{log}_b N$

  • $ext{log}<em>b rac{M}{N} = ext{log}</em>b M - ext{log}_b N$

  • $k ext{ log}<em>b M = ext{log}</em>b (M^k)$

Section 3.4: Solving Exponential and Logarithmic Equations

• Steps to Solve:

  • Isolate the exponential or logarithmic expression first.

  • Apply inverse operations (i.e., log to isolate exponents and vice versa).

  • One-to-One Property:

  • If $a^x = a^y$, then $x = y$

  • Check for extraneous solutions after solving.

Example Problems

1. Graphing:

   • Given $f(x) = 4^x$, identify key points and characteristics, such as intercepts and transformations.

   • For $g(x) = -4^{x-2} - 5$, note:

     • Reflected over the $x$-axis

     • Shift right by 2 and down by 5.

2. Transformations of Logs:

   • $f(x) = ext{log}<em>3 x$ and $g(x) = - ext{log}</em>3(x+3)+4$ demonstrates a reflection over the $x$-axis, a left shift by 3, and an upward shift by 4.

3. Evaluate Expressions:

   • For example, calculate:

     • $ext{log}_{10} ext{√1000}$, apply logarithmic rules.

4. Simplifying Expressions:

   • Example: $ext{log}<em>{25}(2) - ext{log}</em>{25}(4)$ can be condensed to single logarithm expressions.

Solving Exponential and Logarithmic Equations

• Common Scenarios:

  1. Exponential: $2^{x+1} = 16$

  • Solve by writing $16$ as a power of $2$: $2^{x+1} = 2^{4}$, thus $x + 1 = 4$ leads to $x = 3$.

  1. Logarithmic: $ext{log}_2 (x-1) = 3$

  • Convert to exponential form: $x - 1 = 2^3$, then solve for $x$: $x = 9$.

Properties of Logarithms for Expanded Expressions

• Examples:

  • $ext{log}<em>{5}(16xy^2) = 4 ext{ log}</em>{5}(2) + ext{log}<em>{5}(x) + 2 ext{log}</em>{5}(y)$

• For condensing:

  • $3 ext{log}<em>{2}(x) - ext{log}</em>{2}(2) = ext{log}<em>{2}(x^3) - ext{log}</em>{2}(2)$ can be expressed as $ext{log}_{2} rac{x^3}{2}$.

Applications in Real-life Scenarios

• Use of logarithmic functions in financial contexts

• Example: Calculate account balance with continuous compounding.

  • Formula: $A = Pe^{rt}$, with solving for $t$ when finding how long an amount will take to double.

1. Graphing Exponential Functions:

Problem: Given $f(x) = 4^x$, identify key points and characteristics, such as intercepts and transformations.

Explanation of the Problem:
To graph the exponential function $f(x) = 4^x$, we need to determine important features:

• Intercepts: The $y$-intercept occurs when $x = 0$, giving us the point (0, 1).

• Transformations: Consider how the graph is affected by changing parameters. In this case, there are no shifts or reflections, thus it remains upwards as the base is greater than 1.

• Key Points: Other points can be calculated, such as at $x=1$ (resulting in $f(1)=4$) or $x=-1$ (resulting in $f(-1)=\frac{1}{4}$).

2. Analyzing Transformations of Logarithmic Functions:

Problem: For $g(x) = -4^{x-2} - 5$, note transformations.

Explanation of the Problem:
Here, we analyze the function:

• Reflected Over the $x$-axis: The negative sign in front of the function indicates a reflection.

• Transformation: The term $x-2$ indicates a shift to the right by 2 units. The -5 signifies a downward shift by 5 units. Therefore, the graph of this function will appear lower on the graph compared to its parent function.

3. Evaluating Logarithmic Expressions:

Problem: Calculate $ext{log}_{10} ext{√1000}$.

Explanation of the Problem:
To evaluate this expression, we need to:

• Simplify $√1000$: Recognizing that $√{1000} = 10^{3/2}$, we can then rewrite the logarithm:
  $ext{log}_{10} (10^{3/2})$

• Use Logarithmic Properties: Apply the property $ext{log}_{b}(b^x) = x$ which directly gives us $\frac{3}{2}$ as the result.

4. Simplifying Logarithmic Expressions:

Problem: $ext{log}{25}(2) - ext{log}{25}(4)$ can be condensed to a single logarithmic expression.

Explanation of the Problem:
Using the properties of logarithms:

• Apply Logarithmic Difference Property: According to $ext{log}b(M) - ext{log}b(N) = ext{log}b \frac{M}{N}$, we transform our problem to:
  $ext{log}{25} \frac{2}{4}$

• Further Simplification: Reducing $\frac{2}{4}$ yields $\frac{1}{2}$, thus the expression simplifies to:
  $ext{log}{25}\frac{1}{2}$.