What is ln((x^5z^2)/(y^3)) ?

The solution to ln((x^5z^2)/(y^3)) is 5ln(x)+2ln(z)-3ln(y)

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simplify ln((x^5z^2)/(y^3))

simplify

ln (\frac{x ^{5} z ^{2}}{y ^{3}})

5 ln (x) + 2 ln (z) - 3 ln (y)

Solution steps

ln (\frac{x ^{5} z ^{2}}{y ^{3}})

Apply log rule:

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

ln (\frac{x ^{5} z ^{2}}{y ^{3}}) = ln (x^{5} z^{2}) - ln (y^{3})

= ln (x^{5} z^{2}) - ln (y^{3})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

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

= ln (x^{5} z^{2}) - 3 ln (y)

Apply log rule:

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

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

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

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

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

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

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

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

= 5 ln (x) + 2 ln (z) - 3 ln (y)

lo g_{6} (\frac{x ( x ^{2} + 8 )}{x ^{2} - 8})

lo g_{2} (ab)

lo g_{2} (\frac{x ( x ^{2} + 1 )}{x ^{2} - 1})

- \frac{5}{9} lo g_{2} (\frac{5}{9}) - \frac{4}{9} lo g_{2} (\frac{4}{9})

What is ln((x^5z^2)/(y^3)) ?
The solution to ln((x^5z^2)/(y^3)) is 5ln(x)+2ln(z)-3ln(y)