Understanding Logarithms

Apply product, quotient, and power rules for logarithms. Math 30-1.

Lesson 3.5 of Exponents & Logarithms in Math 30-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• What is a Logarithm?. Discover why logarithms were invented and what they actually mean. Learn the definition, the three parts of a log, and why the argument must always be positive.
• Converting Between Forms & Evaluating Logarithms. Master the conversion process between exponential and logarithmic form. Learn to evaluate logs by converting to exponentials and discover the common logarithm shorthand.
• Evaluating Logarithms & Special Rules. Learn how to evaluate logarithms by converting to exponential form. Master the three special rules that aren't on the formula sheet: log of the base, log of 1, and undoing the log.
• Logarithms as Inverses & Finding the Equation. Discover how logarithmic and exponential functions are perfect inverses. Compare their properties side-by-side and learn to find inverse functions by swapping variables and solving.
• Finding Intercepts & Expressing in Terms of y. Find x-intercepts of logarithmic functions by setting y = 0 and converting to exponential form. Learn to express one logarithm in terms of another using substitution and exponent laws.

Inside the lesson — a free preview

The Problem That Created Logarithms

You already know how to solve 2^3 = 8 — the answer is 8. But what if the question is flipped?

2^x = 150

There's no clean exponent that works. This is exactly the problem logarithms were invented to solve. A logarithm is simply a way of asking: what exponent do I need?

The Core Idea. A logarithm answers the question: what power do I raise the base to in order to get this number?

The Definition

Every exponential equation has a logarithmic form, and vice versa. They say the exact same thing — just written differently.

Read y = _b(x) as: "y equals log base b of x".

The Three Parts of a Logarithm

Every logarithm _b(x) = y has three parts:

Three Parts. Base → the small subscript number, b

Argument → the number inside the brackets, x (also called the input)

Value → the result of the logarithm, y (this is the exponent)

The Most Important Thing to Understand

Key Insight. A logarithm IS an exponent. When you find _b(x), you are finding the exponent you must raise b to in order to get x.

Let's see this with three examples:

Example 1. _2(8) = 3 because 2^3 = 8

Asking: "What power do I raise 2 to in order to get 8?" Answer: 3.

Example 2. _3(81) = 4 because 3^4 = 81

Asking: "What power do I raise 3 to in order to get 81?" Answer: 4.

Example 3. _5(1) = 0 because 5^0 = 1

Asking: "What power do I raise 5 to in order to get 1?" Answer: 0.

This works for any valid base: _b(1) = 0 always.

Worked example: Detailed Example 1: Converting to Logarithmic Form

Question: Convert each of the following to logarithmic form.

Part a). 5^3 = 125

Step 1: Identify the parts: base = 5, exponent = 3, result = 125.

Step 2: Write in log form — the exponent becomes the answer:

_5(125) = 3

Verify: 5^3 = 5 × 5 × 5 = 125 ✓

Part b). 3^(-2) = (1)/(9)

Step 1: Identify the parts: base = 3, exponent = −2, result = (1)/(9).

Step 2: Write in log form:

_3((1)/(9)) = -2

Verify: 3^(-2) = (1)/(3^2) = (1)/(9) ✓

Part c). ((2)/(5))^(x^3y) = z

Step 1: Identify the parts: base = (2)/(5), exponent = x^3y, result = z.

Step 2: Write in log form — the exponent x^3y becomes the answer:

_2/5(z) = x^3y

Step 3: If the question asks to isolate for y, divide both sides by x^3:

y = _2/5(z)x^3

Part d). 2t = (3x - 1)^(-3)

Step 1: Identify the parts: base = (3x - 1), exponent = −3, result = 2t.

Note: The power is already isolated on the right side.

Step 2: Write in log form:

_3x-1(2t) = -3

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