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

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

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

simplify

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

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

Solution steps

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

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

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

Apply log rule:

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

ln (6^{3 x}) = 3 x ln (6)

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

lo g_{3} (\frac{9}{x})

lo g_{10} (x) + lo g_{10} (5) - lo g_{10} (y)

3 ln (8) + 2 ln (x)

lo g_{3} (30 x (3 x + 4))

\frac{1}{2} (lo g_{10} (x) + lo g_{10} (y))

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