What is 0.5ln(x^2+1)-4ln(0.5)-0.5(ln(x+4)+ln(x)) ?

The solution to 0.5ln(x^2+1)-4ln(0.5)-0.5(ln(x+4)+ln(x)) is ln((16(x^2+1)^{0.5})/(((x+4)x)^{0.5)})

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simplify 0.5ln(x^2+1)-4ln(0.5)-0.5(ln(x+4)+ln(x))

simplify

0.5 ln (x^{2} + 1) - 4 ln (0.5) - 0.5 (ln (x + 4) + ln (x))

ln (\frac{16 ( x ^{2} + 1 ) ^{0.5}}{( ( x + 4 ) x ) ^{0.5}})

Solution steps

0.5 ln (x^{2} + 1) - 4 ln (0.5) - 0.5 (ln (x + 4) + ln (x))

Apply log rule:

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

0.5 ln (x^{2} + 1) = ln ((x^{2} + 1)^{0.5})

= ln ((x^{2} + 1)^{0.5}) - 4 ln (0.5) - 0.5 (ln (x + 4) + ln (x))

Apply log rule:

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

4 ln (0.5) = ln (0. 5^{4})

= ln ((x^{2} + 1)^{0.5}) - ln (0. 5^{4}) - 0.5 (ln (x + 4) + ln (x))

Apply log rule:

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

ln ((x^{2} + 1)^{0.5}) - ln (0. 5^{4}) = ln (\frac{( x ^{2} + 1 ) ^{0.5}}{0. 5 ^{4}})

= ln (\frac{( x ^{2} + 1 ) ^{0.5}}{0. 5 ^{4}}) - 0.5 (ln (x + 4) + ln (x))

Simplify

\frac{( x ^{2} + 1 ) ^{0.5}}{0. 5 ^{4}} : 16 (x^{2} + 1)^{0.5}

= ln (16 (x^{2} + 1)^{0.5}) - 0.5 (ln (x + 4) + ln (x))

Apply log rule:

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

ln (x + 4) + ln (x) = ln (x (x + 4))

= ln (16 (x^{2} + 1)^{0.5}) - 0.5 ln ((x + 4) x)

Apply log rule:

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

0.5 ln (x (x + 4)) = ln ((x (x + 4))^{0.5})

= ln (16 (x^{2} + 1)^{0.5}) - ln ((x (x + 4))^{0.5})

Apply log rule:

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

ln (16 (x^{2} + 1)^{0.5}) - ln ((x (x + 4))^{0.5}) = ln (\frac{16 ( x ^{2} + 1 ) ^{0.5}}{( x ( x + 4 ) ) ^{0.5}})

= ln (\frac{16 ( x ^{2} + 1 ) ^{0.5}}{( ( x + 4 ) x ) ^{0.5}})

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

ln (\frac{- 4.362}{- 4.732})

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

lo g_{10} (7) + lo g_{10} (18) - lo g_{10} (9)

lo g_{4} (x) - 5 lo g_{4} (y) + 2 lo g_{4} (z)

What is 0.5ln(x^2+1)-4ln(0.5)-0.5(ln(x+4)+ln(x)) ?
The solution to 0.5ln(x^2+1)-4ln(0.5)-0.5(ln(x+4)+ln(x)) is ln((16(x^2+1)^{0.5})/(((x+4)x)^{0.5)})