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

The solution to 1/2 ln(x+5)-2ln(x)+ln(y) is ln((y(x+5)^{1/2})/(x^2))

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

simplify

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

ln (\frac{y ( x + 5 ) ^{\frac{1}{2}}}{x ^{2}})

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Multiply fractions:

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

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

- lo g_{e} (x) - lo g_{e} (y)

- 3 ln (x) + x + x y + y - 3 ln (y)

2 lo g_{10} (2 + 3)

- lo g_{10} (3.8 \cdot 1 0^{- 5})

ln (\frac{3 x ^{5}}{y})

What is 1/2 ln(x+5)-2ln(x)+ln(y) ?
The solution to 1/2 ln(x+5)-2ln(x)+ln(y) is ln((y(x+5)^{1/2})/(x^2))