What is log_{e}(3/4 ln|x^2+2|+1/2 ln|x|) ?

The solution to log_{e}(3/4 ln|x^2+2|+1/2 ln|x|) is ln(ln(|x|^{1/2}|x^2+2|^{3/4}))

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simplify log_{e}(3/4 ln|x^2+2|+1/2 ln|x|)

simplify

lo g_{e} (\frac{3}{4} ln x^{2} + 2 + \frac{1}{2} ln ∣ x ∣)

ln (ln (∣ x ∣^{\frac{1}{2}} x^{2} + 2^{\frac{3}{4}}))

Solution steps

lo g_{e} (\frac{3}{4} ln x^{2} + 2 + \frac{1}{2} ln ∣ x ∣)

Apply log rule:

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

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

= lo g_{e} (ln (x^{2} + 2^{\frac{3}{4}}) + \frac{1}{2} ln ∣ x ∣)

Apply log rule:

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

\frac{1}{2} ln ∣ x ∣ = ln (∣ x ∣^{\frac{1}{2}})

= lo g_{e} (ln (x^{2} + 2^{\frac{3}{4}}) + ln (∣ x ∣^{\frac{1}{2}}))

Apply log rule:

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

ln (x^{2} + 2^{\frac{3}{4}}) + ln (∣ x ∣^{\frac{1}{2}}) = ln (∣ x ∣^{\frac{1}{2}} x^{2} + 2^{\frac{3}{4}})

= lo g_{e} (ln (x^{2} + 2^{\frac{3}{4}} ∣ x ∣^{\frac{1}{2}}))

Apply log rule:

lo g_{e} (x) = ln (x)

= ln (ln (x^{2} + 2^{\frac{3}{4}} ∣ x ∣^{\frac{1}{2}}))

ln (\frac{6}{5}) - ln (\frac{4}{6})

3 ln (\frac{5}{2}) + \frac{9}{10}

- \frac{1}{5} ln (x) - 5 [ln (y) - 3 ln (z)]

\frac{30 ln ( \frac{25}{114} )}{ln ( \frac{79}{114} )}

lo g_{2} (\frac{1}{1000})

What is log_{e}(3/4 ln|x^2+2|+1/2 ln|x|) ?
The solution to log_{e}(3/4 ln|x^2+2|+1/2 ln|x|) is ln(ln(|x|^{1/2}|x^2+2|^{3/4}))