What is 1/2 log_{a}(x)-3log_{a}(y)+6log_{a}(z) ?

The solution to 1/2 log_{a}(x)-3log_{a}(y)+6log_{a}(z) is log_{a}((z^6x^{1/2})/(y^3))

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simplify 1/2 log_{a}(x)-3log_{a}(y)+6log_{a}(z)

simplify

\frac{1}{2} lo g_{a} (x) - 3 lo g_{a} (y) + 6 lo g_{a} (z)

lo g_{a} (\frac{z ^{6} x ^{\frac{1}{2}}}{y ^{3}})

Solution steps

\frac{1}{2} lo g_{a} (x) - 3 lo g_{a} (y) + 6 lo g_{a} (z)

Apply log rule:

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

\frac{1}{2} lo g_{a} (x) = lo g_{a} (x^{\frac{1}{2}})

= lo g_{a} (x^{\frac{1}{2}}) - 3 lo g_{a} (y) + 6 lo g_{a} (z)

Apply log rule:

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

3 lo g_{a} (y) = lo g_{a} (y^{3})

= lo g_{a} (x^{\frac{1}{2}}) - lo g_{a} (y^{3}) + 6 lo g_{a} (z)

Apply log rule:

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

lo g_{a} (x^{\frac{1}{2}}) - lo g_{a} (y^{3}) = lo g_{a} (\frac{x ^{\frac{1}{2}}}{y ^{3}})

= lo g_{a} (\frac{x ^{\frac{1}{2}}}{y ^{3}}) + 6 lo g_{a} (z)

Apply log rule:

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

6 lo g_{a} (z) = lo g_{a} (z^{6})

= lo g_{a} (\frac{x ^{\frac{1}{2}}}{y ^{3}}) + lo g_{a} (z^{6})

Apply log rule:

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

lo g_{a} (\frac{x ^{\frac{1}{2}}}{y ^{3}}) + lo g_{a} (z^{6}) = lo g_{a} (\frac{x ^{\frac{1}{2}}}{y ^{3}} z^{6})

= lo g_{a} (\frac{x ^{\frac{1}{2}}}{y ^{3}} z^{6})

Multiply fractions:

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

= lo g_{a} (\frac{z ^{6} x ^{\frac{1}{2}}}{y ^{3}})

- lo g_{b} (\frac{1}{8})

ln (x) + ln (x - 1)

1 + lo g_{\frac{1}{3}} (x) + lo g_{\frac{1}{3}} (x + 2)

2 lo g_{b} (6) - lo g_{b} (9)

\frac{1}{2} lo g_{10} (x) - 3 lo g_{10} (y) - lo g_{10} (z)

What is 1/2 log_{a}(x)-3log_{a}(y)+6log_{a}(z) ?
The solution to 1/2 log_{a}(x)-3log_{a}(y)+6log_{a}(z) is log_{a}((z^6x^{1/2})/(y^3))