What is 1/2 ln(x^2+9)+ln(3/(sqrt(x^2+9))) ?

The solution to 1/2 ln(x^2+9)+ln(3/(sqrt(x^2+9))) is ln(3)

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simplify 1/2 ln(x^2+9)+ln(3/(sqrt(x^2+9)))

simplify

\frac{1}{2} ln (x^{2} + 9) + ln (\frac{3}{x ^{2} + 9})

ln (3)

Decimal Notation

1.09861 \dots

Solution steps

\frac{1}{2} ln (x^{2} + 9) + ln (\frac{3}{x ^{2} + 9})

Apply log rule:

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

\frac{1}{2} ln (x^{2} + 9) = ln ((x^{2} + 9)^{\frac{1}{2}})

= ln ((x^{2} + 9)^{\frac{1}{2}}) + ln (\frac{3}{x ^{2} + 9})

Apply log rule:

lo g_{c} (a) + lo g_{c} (b) = lo g_{c} (ab)

ln ((x^{2} + 9)^{\frac{1}{2}}) + ln (\frac{3}{x ^{2} + 9}) = ln (\frac{3}{x ^{2} + 9} (x^{2} + 9)^{\frac{1}{2}})

= ln (\frac{3}{x ^{2} + 9} (x^{2} + 9)^{\frac{1}{2}})

(x^{2} + 9)^{\frac{1}{2}} \frac{3}{x ^{2} + 9} = 3

= ln (3)

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What is 1/2 ln(x^2+9)+ln(3/(sqrt(x^2+9))) ?
The solution to 1/2 ln(x^2+9)+ln(3/(sqrt(x^2+9))) is ln(3)