Laws of Logarithms
Edexcel International A Level (IAL) Maths: Pure 2
The Big Idea: Logarithms are the inverse of exponents. Master their laws and you can simplify any logarithmic expression, solve exponential equations, and convert between different bases.
Overview: What You'll Learn
- Exponential functions — what y = ax looks like, how it grows or decays
- Logarithmic functions — what logarithms are and why they're inverses of exponentials
- Three essential laws — how to combine and simplify logarithmic expressions
- Special results — loga(a) = 1, loga(1) = 0, and more
- Change of base formula — how to convert between different bases
- Solving exponential equations — techniques for when x is in the exponent
Exponential Functions
What is an exponential function?
An exponential function has the form y = ax, where a > 0 and a ≠ 1. The key feature is that the variable x is in the exponent (the power), not in the base.
Think of it like compound interest. If you invest £100 at 5% per year, your money multiplies by 1.05 each year. After x years, you have £100 × 1.05x. That's an exponential function.
Key fact: Every exponential graph y = ax passes through the point (0, 1), because a0 = 1 for any positive a. This is where all exponential curves intersect the y-axis.
What does the graph look like?
The shape depends on whether a is greater than 1 or less than 1.
If a > 1 (exponential growth):
The graph curves upward steeply. As x increases, y grows rapidly. As x becomes negative, y approaches zero but never reaches it. The x-axis is a horizontal asymptote—the graph gets closer and closer to the x-axis but never touches it.
Examples: y = 2x, y = 3x, y = 10x
If 0 < a < 1 (exponential decay):
The graph curves downward. As x increases, y decreases toward zero. As x becomes negative, y grows rapidly. Again, the x-axis is an asymptote.
Examples: y = 0.5x, y = 0.3x, y = (1/2)x
Visual trick: y = 0.25x and y = 4x are reflections of each other in the y-axis, because 0.25 = 1/4. So 0.25x = (1/4)x = 4−x.
Worked Example: Comparing Growth Rates
Question: On the same diagram, sketch y = 3x and y = 2x. Which is higher for x > 0?
Step 1: Identify both pass through (0, 1)
Both 30 = 1 and 20 = 1, so both curves pass through (0, 1).
Step 2: Compare growth for x > 0
For x = 1: 31 = 3 vs. 21 = 2. So y = 3x is higher.
For x = 2: 32 = 9 vs. 22 = 4. Still higher.
Rule: When a > 1, the larger the base, the steeper the growth.
Step 3: Compare for x < 0
For x = −1: 3−1 = 1/3 ≈ 0.33 vs. 2−1 = 1/2 = 0.5. Now y = 2x is higher.
Key insight: For negative x, the larger base is "lower" on the graph.
Step 4: Sketch
Both curves pass through (0, 1), stay above the x-axis, and approach it as x → −∞. y = 3x rises faster for x > 0.
Logarithmic Functions
What is a logarithm?
"What power do I raise the base to in order to get this number?"
log3(81)
4
3
x
a