Lesson: Exponential and Logarithmic Functions (Applied Mathematics 1)

Hello, future TCAS students! Today, we’re going to explore the chapter on "Exponential and Logarithmic Functions," which falls under the Number and Algebra strand. This chapter is considered the "heart" of the A-Level exam. Not only does it appear directly in the test, but it is also applied to many other topics, such as compound interest, bacterial growth, and even sound level measurements.

If exponents seem tricky or logarithms look intimidating at first, don’t worry! We’ll break down the content into easy-to-understand pieces, like solving a mystery step-by-step.

1. Essential Prerequisites: Exponents

Before diving into functions, we must master the laws of exponents because they are the vital foundation.

  • Basic Rules: \( a^m \cdot a^n = a^{m+n} \) and \( \frac{a^m}{a^n} = a^{m-n} \)
  • Negative Exponents: \( a^{-n} = \frac{1}{a^n} \) (flip from top to bottom)
  • Roots and Fractions: \( a^{1/n} = \sqrt[n]{a} \)

Important Tip: Don't forget that the base \( a \) cannot be zero when it's in the denominator!

2. Exponential Function

An exponential function is in the form \( f(x) = a^x \), where \( a > 0 \) and \( a \neq 1 \).

Imagine this: If you have 1 Baht and it doubles every day—Day 1 you have 2 Baht, Day 2 you have 4 Baht, Day 3 you have 8 Baht... This rapid growth is exactly what an exponential function represents!

Characteristics of Exponential Graphs

  1. Increasing Function: When the base \( a > 1 \) (the graph slopes upward from left to right).
  2. Decreasing Function: When the base \( 0 < a < 1 \) (the graph slopes downward from left to right).

Note: The graph always passes through the point \( (0, 1) \) (because \( a^0 = 1 \)), and the graph always stays above the \( x \)-axis (the value of \( y \) is always positive).

Did you know? The exponential graph gets closer and closer to the \( x \)-axis but never touches or crosses it. We call this \( x \)-axis the "Asymptote."

3. Solving Exponential Equations and Inequalities

The "Equalize the Bases" Technique

If you have \( a^x = a^y \), you can conclude immediately that \( x = y \).

Example: \( 2^x = 8 \)
Change 8 to base 2, and you get \( 2^x = 2^3 \)
Therefore, \( x = 3 \).

Key point when solving inequalities (Watch out for traps!)
  • If the base \( a > 1 \): The inequality sign "stays the same."
  • If the base \( 0 < a < 1 \): The inequality sign "must be flipped" (e.g., from \( > \) to \( < \)).

Memory trick: "Bases less than 1 (fractions) are troublemakers—you must flip the sign!"

4. Logarithmic Function

A logarithmic function is the "inverse" of an exponential function, written as \( y = \log_a x \).

It means: "To what power must \( a \) be raised to get \( x \)?"

Example: \( \log_2 8 = 3 \) because \( 2^3 = 8 \).

Essential Log Properties (Frequently Tested!)

  1. \( \log_a 1 = 0 \) (because \( a^0 = 1 \))
  2. \( \log_a a = 1 \)
  3. Multiplication becomes addition: \( \log_a (MN) = \log_a M + \log_a N \)
  4. Division becomes subtraction: \( \log_a (\frac{M}{N}) = \log_a M - \log_a N \)
  5. Power Rule (The "kick" rule): \( \log_a M^k = k \log_a M \)
  6. Change of Base: \( \log_a b = \frac{\log_k b}{\log_k a} \)

Common Pitfall: Many students mistakenly think that \( \log(M+N) = \log M + \log N \), which is false! These properties only apply to multiplication and division.

5. Solving Logarithmic Equations and Inequalities

The principle is similar to exponentials: try to make the log bases equal on both sides and then remove the log.

Golden Rule for solving Logs:

"Always check your answers!" Because the value inside the log (the \( x \)) must always be greater than 0, and the base of the log must be greater than 0 and not equal to 1.

Steps to solve:
1. Use log properties to combine terms into a single log on each side.
2. Drop the logs (remember to check the base; if it's less than 1, flip the inequality sign).
3. Solve the equation for \( x \).
4. Check the condition: Ensure the value inside the original log is positive.

6. Real-world Applications

A-Level exam problems often include situational questions related to:

  • Compound Interest: \( A = P(1 + r)^n \) (uses exponentials).
  • Radioactive Decay: Decreasing processes based on half-life.
  • Sound Intensity (Decibels): Uses logs to manage extremely large range of values.

Key Takeaways

  • Exponential: Focus on matching bases and be careful with flipping signs when the base < 1.
  • Logarithm: It’s all about finding the exponent. Master the 5-6 core properties.
  • Crucial Condition: The value inside the log must always be positive! (Never forget to check your answers).
  • Graphs: Memorize the shapes of increasing and decreasing functions to help predict your answers more quickly.

You can do this! This chapter might seem to have many formulas, but if you keep practicing until you can clearly see the relationship between bases, you'll find it's one of the best chapters for scoring well!