Question 724595
You've done some good work so far but I believe there is an error:
{{{y=(1/5)log(3, ((9x-36)^(15))) - 13}}}
{{{y=15*(1/5)log(3, (9x-36)) - 13}}}
{{{y=3log(3, (9x-36)) - 13}}}
{{{y=3log(3, (9(x-4))) - 13}}}
{{{y=3(log(3, (9)) + log(3, (x-4))) - 13}}}
{{{y=3(2 + log(3, (x-4))) - 13}}}
{{{y=6 + 3log(3, (x-4)) - 13}}}
{{{y=-7 + 3log(3, (x-4))}}}
Unless I've made an error, there should be a +3 in front of the log.<br>
The basic function is {{{y = log(3, (x))}}} and its graph looks like:
{{{graph(400, 400, -1, 7, -4, 4, (1/ln(3))*ln(x))}}}
We can see that the domain is all positive numbers (x > 0) and that the range is all real numbers. There is a vertical asymptote at x = 0. The x-intercept is (1, 0).<br>
For {{{y=-7 + 3log(3, (x-4))}}}...<ul><li>The "4" in x-4 will cause a horizontal translation/shift of 4 units to the right.</li><li>The "3" in front of the log will cause a vertical stretch by a factor of 3.</li><li>The -7 will cause a vertical translation/shift of 7 units downward.</li><li>The horizontal shift will shift all x values to the right. So the vertical asymptote and the domain will also be affected in the same way. The new asymptote will be x = 4 and the new domain is x > 4.</li><li>The range of the base function is all real numbers. Vertically shifting and/or stretching all real numbers will not change the range. The range is still all real numbers.</li><li>The new x-intercept is not easily predicted with these transformations. We just have to solve:
{{{0=-7 + 3log(3, (x-4))}}}
{{{7 = 3log(3, (x-4))}}}
{{{7/3 = log(3, (x-4))}}}
{{{3^(7/3) = x - 4}}}
{{{4 + 3^(7/3) = x}}}</li></ul>Here's the graph:
{{{graph(400, 400, -1, 19, -15, 5, -7 + 3*(ln(x-4)/ln(3)))}}}<br>
P.S. In response to the question in your "thank you"...
As you've suspected the "3" in x-3 causes a right shift of three. Both the domain and the asymptote move to the right by 3. The "2" in -2 causes a vertical stretch by  factor of 2. The "-" in -2 causes a reflection in the x-axis. (Think of the graphs above being flipped upside down.) And the -1 is a vertical shift of -1. The range is still all real numbers. To see some y's that are greater than -1, try some x's between 3 and 4.