Find logarithmic equation from a graph but line only clearly passes through one point

Question

I'm working through an online textbook, currently 'graphs of logarithmic functions' on this page, try it #11 right underneath example 11.

For the example before the question, the page shows an example of solving by taking two points on the graph:

This works because there are two clear points on the graph. Then the book provides an opportunity to reinforce the learning with an exercise, #11:

This function only passes through one integer point, (-2,-1). Given that, how can I fins the function using the method I was just shown in example 11?

Answer 1

This question asks for a natural logarithm, whereas the first question asks for a common logarithm. I think that the intent here is that the general form of the logarithm would be $y=\ln(x+b)+c$ and not $y=a\ln(x+b)+c$. The reason is that if the coefficient in front of the $\ln$ is not $1$, then the function could be rewritten as a logarithm with respect to a different base. In particular, the second form could be written as $$ y=\log_{e^a}(x+b)+c. $$ This assumption is not stated in the book (through a simple search), but it makes the problem possible (and provides some ideas for the alternate phrasing). Therefore, we'll consider the form $$ y=\ln(x+b)+c. $$ In this case, it seems that there is enough data in the problem (using the asymptote and the point on the curve).

My solution:

It appears that $x=-3$ is a vertical asymptote, so $b=3$ in our general form and we're interested in $y=\ln(x+3)+c$.

The point $(-2,-1)$ is on the logarithm, so $-1=\ln(-2+3)+c$ or that $-1=\ln(1)+c$, which simplifies to $-1=c$, since $\ln(1)=-1$.