Find equation of log function given graph

Question

How would you find the equation of this log graph with respect to

transformation of the function $$ a\log[k(x-d)] + c$$

I had trouble solving this accurately.

Answer 1

I do not think that you need any transform to find the equation of the graph $$y=a\log[k(x-d)] + c$$ First, you can notice the vertical asymptote at $x=-2$; this implies $d=2$.

Next, $y=0$ if $x=8$; this gives $$0=a \log(10k)+c \implies c=-a \log(10k)$$ So , at this point $$y=a \log[k(x+2)]-a\log(10k)=a \log(\frac {x+2}{10})$$

Now, from the graph, if $x=0$, $y=y_0$ ($\approx -2$) from which you can deduce the value of $$a=-\frac{y_0}{\log (5)}$$ and then $$y=-\frac{y_0}{\log (5)} \log(\frac {x+2}{10})$$ In fact, you could pick any point to fix the value of $a$.

I did not do more since, being almost blind, it is quite hard to me to see accurately the values.

Edit

You could have notice that the equation can write differently since $$y=a\log[k(x-d)] + c=a \log(x-d)+a\log(k)+c=a \log(x-d)+f$$ which simplifies the problem.