Logarithmic Graphs

This lesson covers how logarithmic graphs are used to determine unknown constants in relationships of the form y = axⁿ and y = abˣ. This technique of "reducing to linear form" is a key part of the Edexcel 9MA0 specification and appears regularly in exam papers.

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Why Use Logarithmic Graphs?

When data follows a power law (y = axⁿ) or an exponential law (y = abˣ), plotting the raw data gives a curve. It is difficult to read off the constants a and n (or a and b) from a curve.

By taking logarithms, we can transform the relationship into a straight line of the form Y = mX + c, where the gradient and intercept give us the unknown constants.

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Case 1: y = axⁿ (Power Model)

Take log₁₀ of both sides:

log(y) = log(a × xⁿ) = log(a) + log(xⁿ) = log(a) + n × log(x)

This has the form:

log(y) = n × log(x) + log(a)

Comparing with Y = mX + c:

• Plot log(y) on the vertical axis and log(x) on the horizontal axis
• Gradient = n (the power)
• y-intercept = log(a), so a = 10^(intercept)