What is ln(((3^x6^{2x}))/(2^{2x)}) ?

The solution to ln(((3^x6^{2x}))/(2^{2x)}) is xln(3)+2xln(6)-2xln(2)

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simplify ln(((3^x6^{2x}))/(2^{2x)})

simplify

ln (\frac{( 3 ^{x} 6 ^{2 x} )}{2 ^{2 x}})

x ln (3) + 2 x ln (6) - 2 x ln (2)

Solution steps

ln (\frac{( 3 ^{x} \cdot 6 ^{2 x} )}{2 ^{2 x}})

Apply log rule:

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

ln (\frac{( 3 ^{x} \cdot 6 ^{2 x} )}{2 ^{2 x}}) = ln ((3^{x} \cdot 6^{2 x})) - ln (2^{2 x})

= ln (3^{x} \cdot 6^{2 x}) - ln (2^{2 x})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln (2^{2 x}) = 2 x ln (2)

= ln (3^{x} \cdot 6^{2 x}) - 2 x ln (2)

Apply log rule:

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

ln (3^{x} \cdot 6^{2 x}) = ln (3^{x}) + ln (6^{2 x})

= ln (3^{x}) + ln (6^{2 x}) - 2 ln (2) x

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln (3^{x}) = x ln (3)

= x ln (3) + ln (6^{2 x}) - 2 x ln (2)

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln (6^{2 x}) = 2 x ln (6)

= x ln (3) + 2 x ln (6) - 2 x ln (2)

lo g_{6} (\frac{1}{z ^{3}})

ln (e^{x} x)

- ln (3) + 2 ln (2)

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

ln (4 x)

What is ln(((3^x6^{2x}))/(2^{2x)}) ?
The solution to ln(((3^x6^{2x}))/(2^{2x)}) is xln(3)+2xln(6)-2xln(2)