What is ln(x+3)+2ln(y)-ln(a)+ln(b) ?

The solution to ln(x+3)+2ln(y)-ln(a)+ln(b) is ln((by^2(x+3))/a)

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simplify ln(x+3)+2ln(y)-ln(a)+ln(b)

simplify

ln (x + 3) + 2 ln (y) - ln (a) + ln (b)

ln (\frac{b y ^{2} ( x + 3 )}{a})

Solution steps

ln (x + 3) + 2 ln (y) - ln (a) + ln (b)

Apply log rule:

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

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

= ln (x + 3) + ln (y^{2}) - ln (a) + ln (b)

Apply log rule:

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

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

= ln ((x + 3) y^{2}) - ln (a) + ln (b)

Apply log rule:

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

ln (y^{2} (x + 3)) - ln (a) = ln (\frac{y ^{2} ( x + 3 )}{a})

= ln (\frac{( x + 3 ) y ^{2}}{a}) + ln (b)

Apply log rule:

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

ln (\frac{y ^{2} ( x + 3 )}{a}) + ln (b) = ln (b \frac{y ^{2} ( x + 3 )}{a})

= ln (b \frac{y ^{2} ( x + 3 )}{a})

Multiply fractions:

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

= ln (\frac{b y ^{2} ( x + 3 )}{a})

- lo g_{10} (8 \cdot 1 0^{- 6})

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

ln (\frac{e ^{x}}{2})

2 ln (x + 1) - 2 ln (x) + 4 ln (3 x + 1)

\frac{1}{3} lo g_{10} (m) - 2 lo g_{10} (n)

What is ln(x+3)+2ln(y)-ln(a)+ln(b) ?
The solution to ln(x+3)+2ln(y)-ln(a)+ln(b) is ln((by^2(x+3))/a)