# How to graph and find the x intercept of $h(x)=3\ln(x)-9$

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: $$\begin{array}{c|c} x & y -9 \\ \hline \frac1e & -1-9=-10 \\ 1 & 0-9= -9 \\ e & 1-9= -8 \\ \end{array}$$ 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.

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

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?

• You can obtain $\ln x$ by typing $\ln x$. Sep 8, 2021 at 9:48
• I can. Thanks. For some reason I thought x had to be solved first. Sep 8, 2021 at 10:09
• I see the x intercept is $(e^3, 0)$. Sep 8, 2021 at 15:27

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 $$\ln x = 3$$: • 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. Sep 8, 2021 at 10:11
• I'm using Desmos, but Geogebra also works fine. Sep 8, 2021 at 10:15
• Thanks for the help. I just need to stretch it by 3. Sep 8, 2021 at 14:46