What is 2ln(ln(e^z)ln(e^y))-ln(x^2) ?

The solution to 2ln(ln(e^z)ln(e^y))-ln(x^2) is ln((z^2y^2)/(x^2))

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simplify 2ln(ln(e^z)ln(e^y))-ln(x^2)

simplify

2 ln (ln (e^{z}) ln (e^{y})) - ln (x^{2})

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

Solution steps

2 ln (ln (e^{z}) ln (e^{y})) - ln (x^{2})

Apply log rule:

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

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

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

Apply log rule:

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

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

= ln (\frac{( ln ( e ^{z} ) ln ( e ^{y} ) ) ^{2}}{x ^{2}})

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

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

lo g_{a} (\frac{2}{5})

4.74 + lo g_{10} (\frac{0.2}{0.3})

lo g_{b} (\frac{x ^{3} y}{z ^{7}})

lo g_{2} (\frac{2}{x})

\frac{7}{6} + 2 ln (2) - ln (3) - 5 (ln (6) - ln (5))

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