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How do I sketch the natural logarithm without the use of a calculator/computer? Is there any easy way to find an approximation or do I have to use the repeated integral solution on Wikipedia? I have a test in single-variable analysis and I'm supposed to sketch equations with $x\ln(x)$ for instance.

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1: Find the domain and $x$-intercepts. 2: Find where $f$ is increasing/decreasing. 3: Find where $f$ is concave up/concave down. 4: Determine the end-behaviour of $f(x)$. –  JavaMan Feb 12 '12 at 22:06

4 Answers 4

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Take the derivative (draw tangents)! $$ \frac{d}{dx}\ln x=\frac1x $$ Start with $(1,0)$. Work in both directions. Since you know the slope, draw a short line segment. Then recalculate/readjust the slope at the endpoints of where you have drawn so far. If you need better accuracy, or if you think you are accumulating too much error, read up on Runge-Kutta methods, or find another method to find an absolute reference point. It's definitely easier to keep on track, for example, if you have a table of values $\{e^k\}_{k=-3}^{3}$.

Also, take the second derivative, and graph that first. It is easier to compute, and will be a handy reference (for concavity).

In drawing freehand, it helps to know what you want to draw ahead of time. The logarithm, $y=\ln x$, should definitely be in your repertiore of images, along with the trigonometric functions and $y=e^x$ (and others like $y=x^n$, $y=mx+b$, $(\frac{x}{a})^n+(\frac{y}{b})^n=1$, $y=|x|$, $y=\lfloor x\rfloor$, etc.), which you can draw a rough sketch of from memory.

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Typically, when they ask you to sketch something, they're just asking you to get the major features right. Things like asymptotes, zeros, known special points, increasing/decreasing, maybe even concavity.

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For a sketch of the curve $y=\log_e(x)$, I would expect to see it continuous, passing through $(1,0)$, $(e,1)$ and $(1/e,-1)$ where $e\approx 2.7$, negative when $0 \lt x \lt 1$ and heading towards $-\infty$ as $x$ falls towards $0$ from above, positive for $x \gt 1$, growing very slowly (far below linear) for $x \gt e$, and non-existent for negative $x$.

Knowing the above, for a sketch of $y=x \;\log_e(x)$, I would then say it should also be continuous, passing through $(1,0)$, $(e,e)$ and $(1/e,-1/e)$ where $e\approx 2.7$, negative when $0 \lt x \lt 1$ but closer to zero than $y=\log_e(x)$, positive for $x \gt 1$ and further from zero than $y=\log_e(x)$, growing slightly faster than linear for $x \gt e$, and non-existent for negative $x$. Slightly harder to find, I would also hope to see a minimum at $(1/e,-1/e)$ found by calculus and approaching $(0,0)$ for small positive $x$.

So something like this

enter image description here

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In addition to the points suggested by Hurkyl above, consider the f0llowing as things that could help you plot an accurate graph in general:

1-Is the equation for a function or is it a relation? (understand the definition of a function)

2-What is the valid values of x (domain) where the function is defined or the drawing is required

3-In case of Trig functions, check whether a radian plot or degree plot is required

4-Attempt to check for function roots

5-Attempt to determine the max. and min. values using derivatives

6-Check the behavior (select 2 points for each of the following):

6.A between roots

6.B before roots

6.C after roots

7.Calculate the value for few arbitrary points not covered int the above points

Also, be familiar with some curves such as ln, $e^x$, Sin(x), etc.

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