I have this question on my homework: $$\\f(x)=\ln{(x-3)}\\g(x)=\frac12x-7\\\text{solve for x in:}f(x)=g(x)$$
I have used the substitution property to get this: $$\ln{(x-3)}=\frac12x-7$$.
I don't know how to solve for $x$ from here. Any level of math is ok, I just need to figure out how to solve it. Thanks in advance!
On
From the graph below, two solutions can be roughly approximated : $x\simeq 3$ and $x\simeq 19.5$
Various methods of numerical calculus leads to better approximates : $x\simeq 3.004095$ and $x\simeq 19.6214$
For lovers of special functions : See below the analytical solution, thanks to the Lambert W function, which is a multivaluated function.
Consider the function $$ h(x)=f(x)-g(x)=\ln(x-3)-\frac{x}{2}+7 $$ Then $\lim_{x\to3}h(x)=-\infty$ and $\lim_{x\to\infty}h(x)=-\infty$. Moreover $$ h'(x)=\frac{1}{x-3}-\frac{1}{2}=\frac{5-x}{x-3} $$ so $h$ has a maximum for $x=5$. Now $$ h(5)=\ln2-\frac{5}{2}+7=\ln2+\frac{9}{2}>0 $$ So the equation has two solutions by the intermediate value theorem applied to the intervals $(3,5]$ and $[5,\infty)$.
The solutions can be approximated with several numeric methods.