What is ln(x^{1/2}(x^4)^{1/2}(y^8)^{1/2})-7ln(x) ?

The solution to ln(x^{1/2}(x^4)^{1/2}(y^8)^{1/2})-7ln(x) is ln((y^4)/(x^{9/2)})

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simplify ln(x^{1/2}(x^4)^{1/2}(y^8)^{1/2})-7ln(x)

simplify

ln (x^{\frac{1}{2}} (x^{4})^{\frac{1}{2}} (y^{8})^{\frac{1}{2}}) - 7 ln (x)

ln (\frac{y ^{4}}{x ^{\frac{9}{2}}})

Solution steps

ln (x^{\frac{1}{2}} (x^{4})^{\frac{1}{2}} (y^{8})^{\frac{1}{2}}) - 7 ln (x)

Apply log rule:

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

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

= ln (x^{\frac{1}{2}} (x^{4})^{\frac{1}{2}} (y^{8})^{\frac{1}{2}}) - ln (x^{7})

Apply log rule:

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

ln (x^{\frac{1}{2}} (x^{4})^{\frac{1}{2}} (y^{8})^{\frac{1}{2}}) - ln (x^{7}) = ln \frac{x ^{\frac{1}{2}} ( x ^{4} ) ^{\frac{1}{2}} ( y ^{8} ) ^{\frac{1}{2}}}{x ^{7}}

= ln \frac{x ^{\frac{1}{2}} ( x ^{4} ) ^{\frac{1}{2}} ( y ^{8} ) ^{\frac{1}{2}}}{x ^{7}}

Simplify

\frac{x ^{\frac{1}{2}} ( x ^{4} ) ^{\frac{1}{2}} ( y ^{8} ) ^{\frac{1}{2}}}{x ^{7}} : \frac{y ^{4}}{x ^{\frac{9}{2}}}

= ln (\frac{y ^{4}}{x ^{\frac{9}{2}}})

lo g_{5} (8 x) - lo g_{5} (5 x + 7)

lo g_{2} (2 n)

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

ln (θ e^{- θ x})

4 lo g_{10} (y) - lo g_{10} (x)

What is ln(x^{1/2}(x^4)^{1/2}(y^8)^{1/2})-7ln(x) ?
The solution to ln(x^{1/2}(x^4)^{1/2}(y^8)^{1/2})-7ln(x) is ln((y^4)/(x^{9/2)})