What is 3ln(2)+5ln(x)-1/2 ln(x+7) ?

The solution to 3ln(2)+5ln(x)-1/2 ln(x+7) is ln((8x^5sqrt(x+7))/(x+7))

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simplify 3ln(2)+5ln(x)-1/2 ln(x+7)

simplify

3 ln (2) + 5 ln (x) - \frac{1}{2} ln (x + 7)

ln (\frac{8 x ^{5} x + 7}{x + 7})

Solution steps

3 ln (2) + 5 ln (x) - \frac{1}{2} ln (x + 7)

Apply log rule:

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

3 ln (2) = ln (2^{3})

= ln (2^{3}) + 5 ln (x) - \frac{1}{2} ln (x + 7)

Apply log rule:

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

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

= ln (2^{3}) + ln (x^{5}) - \frac{1}{2} ln (x + 7)

Apply log rule:

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

ln (2^{3}) + ln (x^{5}) = ln (2^{3} x^{5})

= ln (2^{3} x^{5}) - \frac{1}{2} ln (x + 7)

2^{3} = 8

= ln (8 x^{5}) - \frac{1}{2} ln (x + 7)

Apply log rule:

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

\frac{1}{2} ln (x + 7) = ln ((x + 7)^{\frac{1}{2}})

= ln (8 x^{5}) - ln ((x + 7)^{\frac{1}{2}})

Apply log rule:

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

ln (8 x^{5}) - ln ((x + 7)^{\frac{1}{2}}) = ln (\frac{8 x ^{5}}{( x + 7 ) ^{\frac{1}{2}}})

= ln (\frac{8 x ^{5}}{( x + 7 ) ^{\frac{1}{2}}})

Simplify

\frac{8 x ^{5}}{( x + 7 ) ^{\frac{1}{2}}} : \frac{8 x ^{5} x + 7}{x + 7}

= ln (\frac{8 x ^{5} x + 7}{x + 7})

- \frac{3}{4} lo g_{e} (\frac{3}{4}) - \frac{1}{4} lo g_{e} (\frac{1}{4})

ln (41) - ln (5) - ln (2)

lo g_{7} (\frac{ln ( 0.25 )}{ln ( 0.75 )})

2 ln (\frac{308.15}{298.15})

8.31 ln (\frac{1}{10})

What is 3ln(2)+5ln(x)-1/2 ln(x+7) ?
The solution to 3ln(2)+5ln(x)-1/2 ln(x+7) is ln((8x^5sqrt(x+7))/(x+7))