Solve below log Equation, this equation came with different bases:log2(x+2)-3*log8(x+3)=2

Logarithms

Solve below log Equation, this equation came with different bases: $$\displaystyle{\log{{2}}}{\left({x}+{2}\right)}-{3}\cdot{\log{{8}}}{\left({x}+{3}\right)}={2}$$

2020-12-04
$$\displaystyle{\log{{2}}}{\left({x}+{2}\right)}-{3}\cdot{\log{{8}}}{\left({x}+{3}\right)}={2}$$
$$\displaystyle{8}={2}^{{3}},\ {s}{o}\ {\log{{8}}}={\left(\frac{{1}}{{3}}\right)}{\log{{2}}}$$
$$\displaystyle{\log{{2}}}{\left({x}+{2}\right)}-{\log{{2}}}{\left({x}+{3}\right)}={2}$$
$$\displaystyle{\log{{2}}}{\left[\frac{{{x}+{2}}}{{{x}+{3}}}\right]}={2}$$
thus, $$\displaystyle\frac{{{x}+{2}}}{{{x}+{3}}}={2}^{{2}}={4}$$
x+3=4*(x+3)
x+3=4x+12
3x=3-12=-9
x=-3