Year 11 Maths

Logarithmic Functions

Understand the definition of logarithms, master the log laws, and learn to solve logarithmic equations.

What is a Logarithm?

A logarithm is the inverse of an exponential function. It answers the question: "What power must I raise the base to in order to get this number?"

If b^y = x, then log_b(x) = y

Read: "log base b of x equals y"

The exponential and logarithmic forms are equivalent statements. You can convert between them freely:

Exponential Form

2³ = 8

Logarithmic Form

log₂(8) = 3

Common bases: log₁₀(x) is written as log(x) (common log) and log_e(x) is written as ln(x) (natural log), where e ≈ 2.718.

Logarithm Laws

These laws are essential for simplifying and solving logarithmic expressions:

Product Law log_b(MN) = log_b(M) + log_b(N)

Quotient Law log_b(M/N) = log_b(M) - log_b(N)

Power Law log_b(M^k) = k · log_b(M)

Change of Base log_b(x) = log(x)/log(b)

Special Values

log_b(1) = 0

because b⁰ = 1

log_b(b) = 1

because b¹ = b

b^log_b(x) = x

log and exponential undo each other

Solving Logarithmic Equations

Common strategies for solving equations involving logarithms:

Convert to exponential form: If log_b(x) = k, then x = b^k.
Use log laws to combine or simplify multiple log terms into one.
If log_b(A) = log_b(B), then A = B (one-to-one property).
Always check that your answer gives a positive argument for the log (since log of a non-positive number is undefined).

Domain restriction: log_b(x) is only defined for x > 0. Always verify your solutions satisfy this condition.

Key Vocabulary

Logarithm

The inverse of exponentiation. log_b(x) = y means b^y = x.

Common Logarithm

A logarithm with base 10, written as log(x) without a subscript.

Natural Logarithm

A logarithm with base e ≈ 2.718, written as ln(x).

Change of Base

A formula to convert between bases: log_b(x) = log(x) / log(b). Useful for calculator evaluation.

Worked Examples

Evaluate log₃(81).

Step 1: We need to find y such that 3^y = 81.

Step 2: Since 3⁴ = 81, we have y = 4.

Answer: log₃(81) = 4.

Simplify log₂(8) + log₂(4).

Method 1 (Product Law): log₂(8) + log₂(4) = log₂(8 × 4) = log₂(32) = 5 (since 2⁵ = 32).

Method 2 (Direct): log₂(8) = 3 and log₂(4) = 2, so 3 + 2 = 5.

Solve log₅(2x - 1) = 2.

Step 1: Convert to exponential form: 5² = 2x - 1.

Step 2: 25 = 2x - 1, so 2x = 26, thus x = 13.

Step 3: Check: 2(13) - 1 = 25 > 0. ✓ (valid argument)

Answer: x = 13.

Key Concepts Summary

●A logarithm is the inverse of an exponential: if b^y = x, then log_b(x) = y.
●The three key log laws are: product (log of a product = sum of logs), quotient (log of a quotient = difference of logs), and power (log of a power = power times log).
●log_b(1) = 0 and log_b(b) = 1 are fundamental identities.
●The change of base formula log_b(x) = log(x)/log(b) allows calculator evaluation of any base.
●Logarithms are only defined for positive arguments — always check solutions.

Exponential Functions Introduction to Calculus