What is ln(x)-ln(x-4)-1/2 ln(x+1) ?

The solution to ln(x)-ln(x-4)-1/2 ln(x+1) is ln((xsqrt(x+1))/((x-4)(x+1)))

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simplify ln(x)-ln(x-4)-1/2 ln(x+1)

simplify

ln (x) - ln (x - 4) - \frac{1}{2} ln (x + 1)

ln (\frac{x x + 1}{( x - 4 ) ( x + 1 )})

Solution steps

ln (x) - ln (x - 4) - \frac{1}{2} ln (x + 1)

Apply log rule:

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

ln (x) - ln (x - 4) = ln (\frac{x}{x - 4})

= ln (\frac{x}{x - 4}) - \frac{1}{2} ln (x + 1)

Apply log rule:

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

\frac{1}{2} ln (x + 1) = ln ((x + 1)^{\frac{1}{2}})

= ln (\frac{x}{x - 4}) - ln ((x + 1)^{\frac{1}{2}})

Apply log rule:

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

ln (\frac{x}{x - 4}) - ln ((x + 1)^{\frac{1}{2}}) = ln (\frac{\frac{x}{x - 4}}{( x + 1 ) ^{\frac{1}{2}}})

= ln (\frac{\frac{x}{x - 4}}{( x + 1 ) ^{\frac{1}{2}}})

Simplify

\frac{\frac{x}{x - 4}}{( x + 1 ) ^{\frac{1}{2}}} : \frac{x x + 1}{( x - 4 ) ( x + 1 )}

= ln (\frac{x x + 1}{( x - 4 ) ( x + 1 )})

lo g_{x} (\frac{a y ^{4}}{z ^{4}})

ln (\frac{e}{y ^{3}})

- (\frac{9}{14}) lo g_{10} (\frac{9}{14}) - (\frac{5}{14}) lo g_{10} (\frac{5}{14})

ln (\frac{1}{2}) - ln (1)

3 lo g_{10} (9) - 3 lo g_{10} (x)

What is ln(x)-ln(x-4)-1/2 ln(x+1) ?
The solution to ln(x)-ln(x-4)-1/2 ln(x+1) is ln((xsqrt(x+1))/((x-4)(x+1)))