Question

# log (x + 5) - log (x - 3) = log 2

Logarithms
$$\displaystyle{\log{{\left({x}+{5}\right)}}}-{\log{{\left({x}-{3}\right)}}}={\log{{2}}}$$

2020-10-20
We are given: $$\displaystyle{\log{{\left({x}+{5}\right)}}}-{\log{{\left({x}-{3}\right)}}}={\log{{2}}}$$
Use the quotient property of logarithms: $$\displaystyle{\log{{b}}}{x}-{\log{{b}}}{y}={\log{{b}}}{\left(\frac{{x}}{{y}}\right)}$$ $$\displaystyle{\log{{\left(\frac{{{x}+{5}}}{{{x}-{3}}}\right)}}}={\log{{2}}}$$
Equate the arguments: $$\displaystyle\frac{{{x}+{5}}}{{{x}-{3}}}={2}$$
Multiply both sides by x=3:
x+5=2(x-3)
x+5=2x-6
-x=-11
x=11