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

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

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

simplify

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

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

Solution steps

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

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

Apply log rule:

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

ln (2^{3 x}) = 3 x ln (2)

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

f (x) = lo g_{5} (\frac{25}{x + 5})

ln (- 1) - ln (- 3)

lo g_{10} (100000 x x - 3)

lo g_{2} (5) - lo g_{2} (40)

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

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