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I am working on a method to graph logarithm functions. I know the log function is the inverse of the exponent function and I like to convert the log function into this form, $a^y=x$. Then I start making tables to transform the parent function to $h(x)$.

How do I graph this function; $$h(x)=3\ln(x)-9$$ First I re-write the equation into a form I find more understandable. The form is $$e^y=x$$ Then my first table is $$ \begin{array}{c|lcr} x & y \\ \hline \frac1e & -1 \\ 1 & \phantom1 0 \\ e & \phantom1 1 \end{array}$$

The idea of these tables is to get the general shape of the graph by making these tables and then find the $x$ intercept. To draw the general graph of $h(x)=3\ln(x)-9$ is to deal with the minus nine first. I have tried to deal with the equation and graph of the equation in number of ways so far. I don't know that this is the right idea. I may want to multiply by $3$ first but I am not sure. I just needed to get something down to start finding ways to graph these equations. The picture is to show the graph. The table then looks like this: $$\begin{array}{c|c} x & y -9 \\ \hline \frac1e & -1-9=-10 \\ 1 & 0-9= -9 \\ e & 1-9= -8 \\ \end{array}$$ home work notebook

I still am having a hard time finding how these graphs work when $y$ does not equal $-1, 0$, or $1$. It makes it hard to find the $x$ intercept when that happens and to finish drawing the graph.

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  • $\begingroup$ You can obtain $\ln x$ by typing $\ln x$. This tutorial explains how to typeset mathematics on this site. $\endgroup$
    – N. F. Taussig
    Sep 8, 2021 at 9:15

2 Answers 2

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The $x$-intercept occurs when the curve touches the $x$-axis, i.e. when $y=0$. Therefore, finding the $x$-intercept is the same as finding the root of the function $h(x)=3\ln(x)-9$: \begin{align} 3\ln(x) - 9 &= 0 \\[4pt] \ln(x) &= 3 \end{align} Can you do the rest?

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    $\begingroup$ You can obtain $\ln x$ by typing $\ln x$. $\endgroup$
    – N. F. Taussig
    Sep 8, 2021 at 9:48
  • $\begingroup$ I can. Thanks. For some reason I thought x had to be solved first. $\endgroup$
    – Benp404
    Sep 8, 2021 at 10:09
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    $\begingroup$ I see the x intercept is $(e^3, 0)$. $\endgroup$
    – Benp404
    Sep 8, 2021 at 15:27
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Write $h(x)$ as $3 \ln x - 9 = 3(\ln x - 3)$. Then you can plot $\ln x - 3$ first, and then scale the $y$-axis by a factor of $3$.

As mentioned before, the $x$-intercept is when $\ln x = 3$:

enter image description here

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  • $\begingroup$ What graphing software are you using. I tried matplotlob for Python for a while but decided the math is what I wanted to learn and the software has to come after. $\endgroup$
    – Benp404
    Sep 8, 2021 at 10:11
  • $\begingroup$ I'm using Desmos, but Geogebra also works fine. $\endgroup$
    – Toby Mak
    Sep 8, 2021 at 10:15
  • $\begingroup$ Thanks for the help. I just need to stretch it by 3. $\endgroup$
    – Benp404
    Sep 8, 2021 at 14:46

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