What is ln(2)+2ln(x)-ln(9)-5ln(y)-6ln(z-4) ?

The solution to ln(2)+2ln(x)-ln(9)-5ln(y)-6ln(z-4) is ln((2x^2)/(9y^5(z-4)^6))

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simplify ln(2)+2ln(x)-ln(9)-5ln(y)-6ln(z-4)

simplify

ln (2) + 2 ln (x) - ln (9) - 5 ln (y) - 6 ln (z - 4)

ln (\frac{2 x ^{2}}{9 y ^{5} ( z - 4 ) ^{6}})

Solution steps

ln (2) + 2 ln (x) - ln (9) - 5 ln (y) - 6 ln (z - 4)

Apply log rule:

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

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

= ln (2) + ln (x^{2}) - ln (9) - 5 ln (y) - 6 ln (z - 4)

Apply log rule:

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

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

= ln (2 x^{2}) - ln (9) - 5 ln (y) - 6 ln (z - 4)

Apply log rule:

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

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

= ln (\frac{2 x ^{2}}{9}) - 5 ln (y) - 6 ln (z - 4)

Apply log rule:

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

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

= ln (\frac{2 x ^{2}}{9}) - ln (y^{5}) - 6 ln (z - 4)

Apply log rule:

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

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

= ln (\frac{\frac{2 x ^{2}}{9}}{y ^{5}}) - 6 ln (z - 4)

Apply the fraction rule:

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

= ln (\frac{2 x ^{2}}{9 y ^{5}}) - 6 ln (z - 4)

Apply log rule:

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

6 ln (z - 4) = ln ((z - 4)^{6})

= ln (\frac{2 x ^{2}}{9 y ^{5}}) - ln ((z - 4)^{6})

Apply log rule:

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

ln (\frac{2 x ^{2}}{9 y ^{5}}) - ln ((z - 4)^{6}) = ln (\frac{\frac{2 x ^{2}}{9 y ^{5}}}{( z - 4 ) ^{6}})

= ln (\frac{\frac{2 x ^{2}}{9 y ^{5}}}{( z - 4 ) ^{6}})

Apply the fraction rule:

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

= ln (\frac{2 x ^{2}}{9 y ^{5} ( z - 4 ) ^{6}})

ln (\frac{\frac{y}{x}}{\frac{y}{x} + 1})

lo g_{5} (5 x)

ln (\frac{1 - \infty}{1 + \infty})

\frac{1}{4} ln (\frac{3}{7})

ln (\frac{1 + a}{b})

What is ln(2)+2ln(x)-ln(9)-5ln(y)-6ln(z-4) ?
The solution to ln(2)+2ln(x)-ln(9)-5ln(y)-6ln(z-4) is ln((2x^2)/(9y^5(z-4)^6))