erweitern log_{10}(sqrt(((x+4)^5)/(x^3)))

Lösung

erweitern

lo g_{10} \frac{( x + 4 ) ^{5}}{x ^{3}}

\frac{1}{2} lo g_{10} (x^{5} + 20 x^{4} + 160 x^{3} + 640 x^{2} + 1280 x + 1024) - \frac{3}{2} lo g_{10} (x)

Schritte zur Lösung

lo g_{10} \frac{( x + 4 ) ^{5}}{x ^{3}}

\frac{( x + 4 ) ^{5}}{x ^{3}} = \frac{x ^{5} + 20 x ^{4} + 160 x ^{3} + 640 x ^{2} + 1280 x + 1024}{x ^{3}}

= lo g_{10} (\frac{x ^{5} + 20 x ^{4} + 160 x ^{3} + 640 x ^{2} + 1280 x + 1024}{x ^{3}})

Schreibe um

= lo g_{10} ((\frac{x ^{5} + 20 x ^{4} + 160 x ^{3} + 640 x ^{2} + 1280 x + 1024}{x ^{3}})^{\frac{1}{2}})

Wende log Regel an

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

angenommen

x \geq 0

= \frac{1}{2} lo g_{10} (\frac{x ^{5} + 20 x ^{4} + 160 x ^{3} + 640 x ^{2} + 1280 x + 1024}{x ^{3}})

Vereinfache

lo g_{10} (\frac{x ^{5} + 20 x ^{4} + 160 x ^{3} + 640 x ^{2} + 1280 x + 1024}{x ^{3}}) : lo g_{10} (x^{5} + 20 x^{4} + 160 x^{3} + 640 x^{2} + 1280 x + 1024) - 3 lo g_{10} (x)

= \frac{1}{2} (lo g_{10} (x^{5} + 20 x^{4} + 160 x^{3} + 640 x^{2} + 1280 x + 1024) - 3 lo g_{10} (x))

Wende das Distributivgesetz an:

a (b - c) = ab - a c

a = \frac{1}{2}, b = lo g_{10} (x^{5} + 20 x^{4} + 160 x^{3} + 640 x^{2} + 1280 x + 1024), c = 3 lo g_{10} (x)

= \frac{1}{2} lo g_{10} (x^{5} + 20 x^{4} + 160 x^{3} + 640 x^{2} + 1280 x + 1024) - \frac{1}{2} \cdot 3 lo g_{10} (x)

= \frac{1}{2} lo g_{10} (x^{5} + 20 x^{4} + 160 x^{3} + 640 x^{2} + 1280 x + 1024) - 3 \cdot \frac{1}{2} lo g_{10} (x)

3 \cdot \frac{1}{2} lo g_{10} (x) = \frac{3}{2} lo g_{10} (x)

= \frac{1}{2} lo g_{10} (x^{5} + 20 x^{4} + 160 x^{3} + 640 x^{2} + 1280 x + 1024) - \frac{3}{2} lo g_{10} (x)

e x p an d \frac{p ( x )}{( x ^{2} + 1 ) ( x - 3 )}

e x p an d (k + 1)!

e x p an d (1 - cos^{2} (θ))^{3}

s im pl i f y (x + 12) (x - \frac{7}{5})

e x p an d x (1 - cos (x))