You can put this solution on YOUR website!
I'm not sure what you mean by "key points". The only ones I can think of would be the x and y intercepts, if any. The x-intercept is the point(s) where a graph crosses/touches/intersects the x-axis. Since all the points on the x-axis have a y coordinate of zero, that is how you find them. You make y zero and solve for x:
Adding the log to both sides we get:
Rewriting this in exponential form we get:
which simplifies to:
Subtracting 5 we get:
Dividing by 2 we get:
So the x-intercept is (3/2, 0)
For the y-intercept we make x zero:
which simplifies to:
Since our calculators can't "do" base 2 logarithms we need to change the base of the logarithm using the conversion formula, , to convert to a base our calculator "knows" (like base 10 or base e (ln)). I'll use base 10: which I'll round off to 0.7. So the y-intercept is near (0, 0.7)
Another item which can be helpful when graphing is the vertical asymptote. The arguments of all logarithms must be positive. So the domain of any equation like yours must have a domain which makes your argument (2x+5) positive. In other "words":
Solving this we get:
So the domain is all numbers greater than -5/2.
We find the domain because the "edge" of the domain will be a vertical asymptote. Just make an equation out of the domain:
x = -5/2
and you have your vertical asymptote!
Two points and an asymptote are not much to go on. You may need other points. To find other points just pick numbers for x or y and use the equation to find the value of other variable (just like we did when we picked 0 for x and found the y-intercept and when we picked 0 for y and found the x-intercept). I'll leave it up to you to find more points.
Algebra.com's graphing feature is not perfect but here's how the graph should look (roughly):
(