What is ln(1/2 |x+sqrt(4+x^2)|) ?

The solution to ln(1/2 |x+sqrt(4+x^2)|) is ln|x+sqrt(x^2+4)|-ln(2)

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simplify ln(1/2 |x+sqrt(4+x^2)|)

simplify

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

ln x + x^{2} + 4 - ln (2)

Solution steps

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

Apply log rule:

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

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

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

Apply log rule:

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

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

= ln (1) - ln (2) + ln x + x^{2} + 4

Apply log rule:

lo g_{a} (1) = 0

= 0 - ln (2) + ln x + x^{2} + 4

0 - ln (2) + ln x + 4 + x^{2} = - ln (2) + ln x + 4 + x^{2}

= ln x + x^{2} + 4 - ln (2)

lo g_{4} (x + 3) + lo g_{4} (x - 3)

lo g_{a} (\frac{x ^{\frac{1}{2}} y ^{5}}{x ^{3}})

lo g_{10} (\frac{p ^{6}}{q ^{1}})

lo g_{10} (12 x^{2})

20 lo g_{10} (\frac{4 π}{\frac{1}{8}}) - 20 lo g_{10} (0.5)

What is ln(1/2 |x+sqrt(4+x^2)|) ?
The solution to ln(1/2 |x+sqrt(4+x^2)|) is ln|x+sqrt(x^2+4)|-ln(2)