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Topic 4: Logarithmic Functions

4 dot points, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

How do we solve an equation when the unknown is in the exponent?

Exponential equations are solved by taking logarithms of both sides and applying the power law to bring the unknown exponent down.

How to solve exponential equations by taking logs and using the power law, with worked growth-and-decay applications including doubling time and half-life.

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What does the graph of a logarithmic function look like and how is it related to the exponential graph?

The logarithmic graph is the reflection of the exponential graph in the line y = x, with a vertical asymptote and a characteristic slow growth.

The shape, domain, asymptote and intercepts of y = log_a(x), how it reflects the exponential graph, and how transformations shift and stretch it.

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How do the laws of logarithms let us simplify and rearrange logarithmic expressions?

The logarithm laws turn products into sums, quotients into differences, and powers into multipliers, mirroring the index laws.

The product, quotient and power laws of logarithms, how they follow from the index laws, the change-of-base rule, and how to use them to simplify and solve.

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How do we differentiate exponential and logarithmic functions?

The derivative of e^x is itself, the derivative of ln x is 1/x, and the chain rule extends both to composite functions.

The derivatives of the exponential and natural log functions, the chain-rule extensions for composite exponentials and logarithms, and worked applications including growth models and stationary points.

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