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Natural Log Mirrors Exponential
Age 17 · Maths
The natural logarithm ln is the inverse of the exponential function e^x. Its graph is the mirror image of the graph of e^x across the line y = x, so coordinates swap places.
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In the picture
- The blue rising curve: exponential growth, climbing faster as x increases
- The orange curve: the natural logarithm graph, falling near the y-axis and then rising slowly
- The dashed diagonal line: the line y = x
- The black dot on the horizontal axis: the point (1, 0), where ln x is zero
Examples
- Continuous compound interest: e^x gives the balance after x years, and ln finds how long it takes to reach a target balance.
- Radioactive decay: e^{-kt} gives the remaining amount, and ln finds the elapsed time from that amount.
Key takeaways
- ln and e^x undo each other.
- Their graphs are reflections of each other across y = x.
Quiz: Natural Log Mirrors Exponential (Age 17)
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What is the relationship between ln x and e^x?
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How is the graph of y = ln x related to the graph of y = e^x?
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If a point is (a, b) on the graph of y = e^x, what point corresponds to it on the graph of y = ln x?
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