04 61loggraphse

# Logarithm Graphs (Base e)

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Recall that $\log_e(x)$ is often written as **ln(x)** {Natural Logarithm}

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The graph of **y = ln(x)** follows the same rule as all other log graphs.

- Asymptote: $x = 0$

- x-intercept: $(1, \; 0)$

- 2nd point: $(e, \; 1)$

- Domain: $x \in R^+$

- Range: $y \in R$

- Strictly Increasing Graph

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## Comparing the natural log graph to y = e^{x}

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Recall that a log graph is the inverse of the exponential graph with the same base.

Therefore the natural log graph $y = \log_e(x)$ is the inverse of the $y = e^x$ graph.

- This means that it is a reflection across the line $y = x$
- or that the
**x**and**y**coordinates of each individual point are swapped.- $(x, \; y) \rightarrow (y, \; x)$

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## Transformations on the Standard Natural Log Graph

We can apply the usual transformations such as dilations, translations and reflections when sketching the natural log graph.

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