the ln of a variable

Question

I have the equation $2t-5.682511886=-2*\ln(3t)+\ln(30t)$

I need to solve this equation to find t but I am unable to do that because of the natural logs

So my question is how do you deal with something like $\ln(6t^2)$ (as an example).

Answer 1

Consider the equation $$2t-a=-2*\log(3t)+\log(30t)=-2\log(3)-2\log(t)+\log(30)+\log(t)$$ that is to say $$2t+\log(t)=b\qquad \text{with} \qquad b=a-2\log(3)+\log(30)$$

The solution for $t$ is given in terms of Lambert function and write $$t=\frac{1}{2} W\left(2 e^b\right)$$

Now, using the value of $a$, $b=6.886484690$ and $t=2.909288133$.

If you cannot use Lambert function, only numerical methods will solve the problem.