What is ln(x-3)-2ln(x+12)-ln(x) ?

The solution to ln(x-3)-2ln(x+12)-ln(x) is ln((x-3)/(x(x+12)^2))

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simplify ln(x-3)-2ln(x+12)-ln(x)

simplify

ln (x - 3) - 2 ln (x + 12) - ln (x)

ln (\frac{x - 3}{x ( x + 12 ) ^{2}})

Solution steps

ln (x - 3) - 2 ln (x + 12) - ln (x)

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

2 ln (x + 12) = ln ((x + 12)^{2})

= ln (x - 3) - ln ((x + 12)^{2}) - ln (x)

Apply log rule:

lo g_{c} (a) - lo g_{c} (b) = lo g_{c} (\frac{a}{b})

ln (x - 3) - ln ((x + 12)^{2}) = ln (\frac{x - 3}{( x + 12 ) ^{2}})

= ln (\frac{x - 3}{( x + 12 ) ^{2}}) - ln (x)

Apply log rule:

lo g_{c} (a) - lo g_{c} (b) = lo g_{c} (\frac{a}{b})

ln (\frac{x - 3}{( x + 12 ) ^{2}}) - ln (x) = ln (\frac{\frac{x - 3}{( x + 12 ) ^{2}}}{x})

= ln (\frac{\frac{x - 3}{( x + 12 ) ^{2}}}{x})

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= ln (\frac{x - 3}{x ( x + 12 ) ^{2}})

ln (- e^{- y} (y + 1))

- ln (\frac{80}{81})

ln (\frac{52}{7})

ln (3 x) - ln (u)

- \frac{1}{4} ln (\frac{1}{3})

What is ln(x-3)-2ln(x+12)-ln(x) ?
The solution to ln(x-3)-2ln(x+12)-ln(x) is ln((x-3)/(x(x+12)^2))