What is 2ln(2)+5ln(x)-1/2 ln(x+9) ?

The solution to 2ln(2)+5ln(x)-1/2 ln(x+9) is ln((4x^5sqrt(x+9))/(x+9))

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simplify 2ln(2)+5ln(x)-1/2 ln(x+9)

simplify

2 ln (2) + 5 ln (x) - \frac{1}{2} ln (x + 9)

ln (\frac{4 x ^{5} x + 9}{x + 9})

Solution steps

2 ln (2) + 5 ln (x) - \frac{1}{2} ln (x + 9)

Apply log rule:

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

2 ln (2) = ln (2^{2})

= ln (2^{2}) + 5 ln (x) - \frac{1}{2} ln (x + 9)

Apply log rule:

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

5 ln (x) = ln (x^{5})

= ln (2^{2}) + ln (x^{5}) - \frac{1}{2} ln (x + 9)

Apply log rule:

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

ln (2^{2}) + ln (x^{5}) = ln (2^{2} x^{5})

= ln (2^{2} x^{5}) - \frac{1}{2} ln (x + 9)

2^{2} = 4

= ln (4 x^{5}) - \frac{1}{2} ln (x + 9)

Apply log rule:

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

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

= ln (4 x^{5}) - ln ((x + 9)^{\frac{1}{2}})

Apply log rule:

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

ln (4 x^{5}) - ln ((x + 9)^{\frac{1}{2}}) = ln (\frac{4 x ^{5}}{( x + 9 ) ^{\frac{1}{2}}})

= ln (\frac{4 x ^{5}}{( x + 9 ) ^{\frac{1}{2}}})

Simplify

\frac{4 x ^{5}}{( x + 9 ) ^{\frac{1}{2}}} : \frac{4 x ^{5} x + 9}{x + 9}

= ln (\frac{4 x ^{5} x + 9}{x + 9})

lo g_{7} (18) - lo g_{7} (6)

lo g_{4} (3) + 2 lo g_{4} (x) + 3 lo g_{4} (y)

3 lo g_{x} (2) + 2 lo g_{x} (3)

6 lo g_{b} (x) + 9 lo g_{b} (z)

lo g_{2} (4 x^{2})

What is 2ln(2)+5ln(x)-1/2 ln(x+9) ?
The solution to 2ln(2)+5ln(x)-1/2 ln(x+9) is ln((4x^5sqrt(x+9))/(x+9))