So I was again looking through the homepage of Youtube to see if there were any math video that I thought that I might be able to solve when I came across this video by Learncommunolizer asking for the value of $x$ in $$\color{black}{9^x=18}$$which I thought I might be able solve. Here is my attempt at solving the aforementioned question:$$9^x=18$$$$3^{2x}=18$$$$3^{2x-1}=6$$$$3^{2x-2}=2$$$$(2x-2)\ln3=\ln2$$$$2x-2=\dfrac{\ln2}{\ln3}$$$$2x=\dfrac{\ln2}{\ln3}+2$$And knowing that$$\dfrac{a}{b}+\dfrac{c}{d}=\dfrac{ad+bc}{cd}$$Also$$2=\dfrac{2}{1}$$$$2x=\dfrac{\ln(2)+2\ln(3)}{\ln(3)}$$$$\therefore x=\dfrac{\ln(2)+2\ln(3)}{2\ln(3)}$$My question
Is the solution that I have achieved correct, or what could I do to achieve the correct solution/attain it more quickly?
You can directly compute
$$ x \ln (9) = \ln(18) $$ $$\Longrightarrow x = \frac{\ln (18)}{\ln (9)}$$
Factoring and using logarithm rules we get
$$x = \frac{\ln (2 \cdot 3^2)}{\ln (3^2)}$$ $$\Longrightarrow x = \frac{\ln (2) + 2\ln(3)}{2\ln (3)}$$
So your result is correct, but with a few extra steps in between.