Step 1 Given equation

$\mathrm{log}(x+2)-\mathrm{log}x=2\mathrm{log}4$

Step 2 Formula

$\mathrm{log}a-\mathrm{log}b=\frac{\mathrm{log}a}{\mathrm{log}b}$

$\mathrm{log}\left(ab\right)=\mathrm{log}a\times \mathrm{log}b$

${\mathrm{log}x}^{n}=n\mathrm{log}x$

Step 3 We need to solve the equation

$\mathrm{log}(x+2)-\mathrm{log}x=2\mathrm{log}4$

$\frac{\mathrm{log}(x+2)}{\mathrm{log}x}=2\mathrm{log}4$

$\frac{\mathrm{log}(x+2)}{\mathrm{log}x}={\mathrm{log}4}^{2}$

Taking anti−log on both sides

$\frac{x+2}{x}=16$

x+2=16x

16-x=2

15x=2

$x=\frac{2}{15}$

Step 4 Answer

$x=\frac{2}{15}$