What is 2ln(ln(x))-ln(ln(x^2)+1) ?

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

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simplify 2ln(ln(x))-ln(ln(x^2)+1)

simplify

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

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

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

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

Apply log rule

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

assuming

x \geq 0

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

ln (\frac{e ^{\frac{0.8}{x}} - 1}{e ^{\frac{0.4}{x}} - 1})

8 ln 2 + 4 - 4 + 2 ln 2 + 4 + 1

lo g_{10} (4 5^{\circ} 1 2^{\circ} 9^{\circ})

- ln ∣ 2 x - 2 ∣ + ln ∣ 2 x - 4 ∣

ln (\frac{x y ^{4}}{z ^{3}})

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