If there's a natural log of two terms, which I cannot simplify with the laws of logarithms, how should I simplify it?
e.g. $\ln(e^{6x} + 17)$
The full equation could be something like this: $$\ln(e^{2x}) + \ln(e^{6x} +17) = \ln(50).$$
I know that I can simplify $\ln(e^{2x})$ to just $2x,$ and I can evaluate $\ln(50)$ with a calculator. But I'm not sure how to simplify the $\ln(e^{6x} + 17).$
$$\ln(e^{2x})+\ln(e^6x+17)=\ln(e^{2x}(e^{6x}+17))=\ln(50) \Rightarrow \\ t(t^3+17)=50 \Rightarrow t=2\\ e^{2x}=2 \Rightarrow x=0.5\ln 2.$$
$$\ln (e^{2x})+\ln (e^{6x}+17) = \ln 50$$ As you know, $$\log_a b+\log_a c = \log_a (bc)$$
$$\ln\big[(e^{2x})(e^{6x}+17)\big] = \ln 50$$ Remove the log from both sides.
$$(e^{2x})(e^{6x}+17) = 50$$ Now, replace $e^{2x}$ with a variable so it can be solved easily. $$m = e^{2x}$$ $$m(m^3+17) = 50$$ $$m = 2$$ $$e^{2x} = 2$$ $$2x = \ln 2 \implies \boxed {x = \frac{\ln 2}{2}}$$
