What is ln((3^x)/((1+x^3)^2)) ?

The solution to ln((3^x)/((1+x^3)^2)) is xln(3)-2ln(1+x^3)

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simplify ln((3^x)/((1+x^3)^2))

simplify

ln (\frac{3 ^{x}}{( 1 + x ^{3} ) ^{2}})

x ln (3) - 2 ln (1 + x^{3})

Solution steps

ln (\frac{3 ^{x}}{( 1 + x ^{3} ) ^{2}})

Apply log rule:

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

ln (\frac{3 ^{x}}{( 1 + x ^{3} ) ^{2}}) = ln (3^{x}) - ln ((1 + x^{3})^{2})

= ln (3^{x}) - ln ((x^{3} + 1)^{2})

Apply log rule:

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

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

= x ln (3) - ln ((1 + x^{3})^{2})

Apply log rule:

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

ln ((1 + x^{3})^{2}) = 2 ln (1 + x^{3})

= x ln (3) - 2 ln (1 + x^{3})

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

(\frac{1}{2 x})^{2} + ln (\frac{1}{2 x})

lo g_{e} (\frac{2117}{1000})

- \frac{1}{2} ln (5)

ln (\frac{x}{100})

What is ln((3^x)/((1+x^3)^2)) ?
The solution to ln((3^x)/((1+x^3)^2)) is xln(3)-2ln(1+x^3)