What is 1/3 ln(x-3)-4ln(z)+ln(y+7) ?

The solution to 1/3 ln(x-3)-4ln(z)+ln(y+7) is ln(((y+7)(x-3)^{1/3})/(z^4))

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simplify 1/3 ln(x-3)-4ln(z)+ln(y+7)

simplify

\frac{1}{3} ln (x - 3) - 4 ln (z) + ln (y + 7)

ln (\frac{( y + 7 ) ( x - 3 ) ^{\frac{1}{3}}}{z ^{4}})

Solution steps

\frac{1}{3} ln (x - 3) - 4 ln (z) + ln (y + 7)

Apply log rule:

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

\frac{1}{3} ln (x - 3) = ln ((x - 3)^{\frac{1}{3}})

= ln ((x - 3)^{\frac{1}{3}}) - 4 ln (z) + ln (y + 7)

Apply log rule:

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

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

= ln ((x - 3)^{\frac{1}{3}}) - ln (z^{4}) + ln (y + 7)

Apply log rule:

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

ln ((x - 3)^{\frac{1}{3}}) - ln (z^{4}) = ln (\frac{( x - 3 ) ^{\frac{1}{3}}}{z ^{4}})

= ln (\frac{( x - 3 ) ^{\frac{1}{3}}}{z ^{4}}) + ln (y + 7)

Apply log rule:

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

ln (\frac{( x - 3 ) ^{\frac{1}{3}}}{z ^{4}}) + ln (y + 7) = ln (\frac{( x - 3 ) ^{\frac{1}{3}}}{z ^{4}} (y + 7))

= ln (\frac{( x - 3 ) ^{\frac{1}{3}}}{z ^{4}} (y + 7))

Multiply fractions:

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

= ln (\frac{( y + 7 ) ( x - 3 ) ^{\frac{1}{3}}}{z ^{4}})

lo g_{10} ((\frac{p ^{4}}{q ^{7}})^{2})

- lo g_{10} (2.6 \cdot 1 0^{- 4})

lo g_{x} (\frac{1}{16})

3 ln (x) - 4 ln (y) - 5 ln (z)

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

What is 1/3 ln(x-3)-4ln(z)+ln(y+7) ?
The solution to 1/3 ln(x-3)-4ln(z)+ln(y+7) is ln(((y+7)(x-3)^{1/3})/(z^4))