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Logarithm mirrors the exponential
Age 18 · Maths
The natural logarithm undoes the exponential function: if y = eˣ, then x = ln y. Their graphs are reflections of each other across the line y = x.
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In the picture
- The exponential curve y = eˣ: rises ever more steeply
- The logarithm curve y = ln x: rises ever more slowly
- The line y = x: the mirror between the two graphs
- The point (0, 1) on the exponential curve: e⁰ = 1
- The point (1, 0) on the logarithm curve: ln 1 = 0, the mirror of (0, 1)
Examples
- Compound growth: eˣ gives the amount after time x, and ln finds the time needed to reach a target amount.
- Sound and earthquake scales: a logarithm turns huge ratios into small, manageable numbers.
Key takeaways
- Swapping x and y turns y = eˣ into y = ln x.
- The domain of ln is x > 0, because eˣ is always positive.
Quiz: Logarithm mirrors the exponential (Age 18)
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If you swap x and y in y = eˣ, which equation do you get?
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For which values of x is ln x defined?
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Across which line are the graphs of eˣ and ln x reflections of each other?
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