What is (1/3 ln|e^x+1|+2/3 ln|e^x-2|)*e^x ?

The solution to (1/3 ln|e^x+1|+2/3 ln|e^x-2|)*e^x is ln(|e^x+1|^{1/3}|e^x-2|^{2/3})e^x

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simplify (1/3 ln|e^x+1|+2/3 ln|e^x-2|)*e^x

simplify

(\frac{1}{3} ln ∣ e^{x} + 1 ∣ + \frac{2}{3} ln ∣ e^{x} - 2 ∣) \cdot e^{x}

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

Solution steps

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

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

\frac{1}{3} ln ∣ e^{x} + 1 ∣ = ln (∣ e^{x} + 1 ∣^{\frac{1}{3}})

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

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

\frac{2}{3} ln ∣ e^{x} - 2 ∣ = ln (∣ e^{x} - 2 ∣^{\frac{2}{3}})

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

Apply log rule:

lo g_{c} (a) + lo g_{c} (b) = lo g_{c} (ab)

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

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

- \frac{1}{0.43861} ln (\frac{5}{1205})

ln (5 \cdot e^{7 - x})

ln (2 x) + ln (x)

ln (6) + ln (\frac{e ^{(1.86367)}}{3})

What is (1/3 ln|e^x+1|+2/3 ln|e^x-2|)*e^x ?
The solution to (1/3 ln|e^x+1|+2/3 ln|e^x-2|)*e^x is ln(|e^x+1|^{1/3}|e^x-2|^{2/3})e^x