Logarithms and Laws

This lesson introduces logarithms and the three key logarithm laws that you need for the Edexcel 9MA0 specification. Logarithms are the inverse of exponential functions and are essential for solving exponential equations.

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What Is a Logarithm?

A logarithm answers the question: "What power must I raise the base to in order to get this number?"

The statement aˣ = b is equivalent to x = log_a(b)

Read "log_a(b)" as "log base a of b."

Examples:

• 2³ = 8, so log₂(8) = 3
• 10² = 100, so log₁₀(100) = 2
• 5⁰ = 1, so log₅(1) = 0

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Important Special Cases

Statement	Explanation
log_a(1) = 0	Because a⁰ = 1 for any base a
log_a(a) = 1	Because a¹ = a
log_a(aⁿ) = n	Applying the definition directly
a^(log_a(x)) = x	The exponential and logarithm cancel

These identities follow directly from the definition and are used constantly.

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The Three Logarithm Laws

Law 1: The Addition Law

log_a(x) + log_a(y) = log_a(xy)

The log of a product equals the sum of the logs.