What is 5ln(z)-3ln(x)-7ln(y) ?

The solution to 5ln(z)-3ln(x)-7ln(y) is ln((z^5)/(x^3y^7))

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simplify 5ln(z)-3ln(x)-7ln(y)

simplify

5 ln (z) - 3 ln (x) - 7 ln (y)

ln (\frac{z ^{5}}{x ^{3} y ^{7}})

Solution steps

5 ln (z) - 3 ln (x) - 7 ln (y)

Apply log rule:

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

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

= ln (z^{5}) - 3 ln (x) - 7 ln (y)

Apply log rule:

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

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

= ln (z^{5}) - ln (x^{3}) - 7 ln (y)

Apply log rule:

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

ln (z^{5}) - ln (x^{3}) = ln (\frac{z ^{5}}{x ^{3}})

= ln (\frac{z ^{5}}{x ^{3}}) - 7 ln (y)

Apply log rule:

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

7 ln (y) = ln (y^{7})

= ln (\frac{z ^{5}}{x ^{3}}) - ln (y^{7})

Apply log rule:

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

ln (\frac{z ^{5}}{x ^{3}}) - ln (y^{7}) = ln (\frac{\frac{z ^{5}}{x ^{3}}}{y ^{7}})

= ln (\frac{\frac{z ^{5}}{x ^{3}}}{y ^{7}})

Apply the fraction rule:

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

= ln (\frac{z ^{5}}{x ^{3} y ^{7}})

lo g_{5} (3 x + 1) + lo g_{5} (x)

lo g_{10} (8) + lo g_{10} (3)

lo g_{3} (12) + lo g_{3} (7) + 4 lo g_{3} (5)

- lo g_{10} (4 \cdot 1 0^{- 8})

lo g_{10} (5) - lo g_{10} (2)

What is 5ln(z)-3ln(x)-7ln(y) ?
The solution to 5ln(z)-3ln(x)-7ln(y) is ln((z^5)/(x^3y^7))