What is ln((3^x6^{3x})/(2^{4x)}) ?

The solution to ln((3^x6^{3x})/(2^{4x)}) is xln(3)+3xln(6)-4xln(2)

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simplify ln((3^x6^{3x})/(2^{4x)})

simplify

ln (\frac{3 ^{x} 6 ^{3 x}}{2 ^{4 x}})

x ln (3) + 3 x ln (6) - 4 x ln (2)

Solution steps

ln (\frac{3 ^{x} \cdot 6 ^{3 x}}{2 ^{4 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 ^{4 x}}) = ln (3^{x} \cdot 6^{3 x}) - ln (2^{4 x})

= ln (3^{x} \cdot 6^{3 x}) - ln (2^{4 x})

Apply log rule:

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

ln (2^{4 x}) = 4 x ln (2)

= ln (3^{x} \cdot 6^{3 x}) - 4 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}) - 4 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}) - 4 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) - 4 x ln (2)

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

ln (\frac{2.5}{2})

lo g_{2} (\frac{p}{q})

f (r) = \frac{1}{2} (lo g_{6} (r - 1) - lo g_{6} (r))

lo g_{10} (\frac{a ^{4} b ^{3}}{c ^{2}})

What is ln((3^x6^{3x})/(2^{4x)}) ?
The solution to ln((3^x6^{3x})/(2^{4x)}) is xln(3)+3xln(6)-4xln(2)