What is ln((8x)^{1/2})+ln(4)x^2-ln((16x)^{1/2}) ?

The solution to ln((8x)^{1/2})+ln(4)x^2-ln((16x)^{1/2}) is -1/2 ln(2)+2ln(2)x^2

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simplify ln((8x)^{1/2})+ln(4)x^2-ln((16x)^{1/2})

simplify

ln ((8 x)^{\frac{1}{2}}) + ln (4) x^{2} - ln ((16 x)^{\frac{1}{2}})

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

Solution steps

ln ((8 x)^{\frac{1}{2}}) + ln (4) x^{2} - ln ((16 x)^{\frac{1}{2}})

Apply log rule:

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

ln ((8 x)^{\frac{1}{2}}) - ln ((16 x)^{\frac{1}{2}}) = ln (\frac{( 8 x ) ^{\frac{1}{2}}}{( 16 x ) ^{\frac{1}{2}}})

= ln (\frac{( 8 x ) ^{\frac{1}{2}}}{( 16 x ) ^{\frac{1}{2}}}) + ln (4) x^{2}

\frac{( 8 x ) ^{\frac{1}{2}}}{( 16 x ) ^{\frac{1}{2}}} = \frac{1}{2 ^{\frac{1}{2}}}

= ln (\frac{1}{2 ^{\frac{1}{2}}}) + ln (4) x^{2}

ln (\frac{1}{2 ^{\frac{1}{2}}}) = - \frac{1}{2} ln (2)

= - \frac{1}{2} ln (2) + ln (4) x^{2}

Simplify

ln (4) : 2 ln (2)

= - \frac{1}{2} ln (2) + 2 ln (2) x^{2}

lo g_{10} (\frac{x + y}{z})

ln (\frac{2 x ^{3}}{( x + 4 ) ^{2}})

lo g_{a} (u^{7} v^{3})

lo g_{y} (+ 11 lo g_{10} (z))

2 ln (40) - 2 ln (20)

What is ln((8x)^{1/2})+ln(4)x^2-ln((16x)^{1/2}) ?
The solution to ln((8x)^{1/2})+ln(4)x^2-ln((16x)^{1/2}) is -1/2 ln(2)+2ln(2)x^2