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)

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

ln (2 n^{2} + 1) - ln (n^{2} + 1)

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

lo g_{4} (w) + 2 lo g_{4} (z)

lo g_{10} (8) - lo g_{10} (125)

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