Logarithmus-Terme vereinfachen

Question

Wie vereinfache ich diese Logarithmus-Terme?

1) \( \pi^{\log _{\pi}\left((x-1)^{2}\right)} \)

2) \( \ln \left(y^{4}\right)+4 \ln \left(y^{-1}\right) \)

3) \( \sqrt{\pi} \log _{\pi}(16) \)

Answer 1

Aufgabe 2)

=ln(y^4) +ln(y^(-4))

=ln( y^4 *y^(-4)) =ln (1)=0

Answer 2

1) = (x-1)^2

3) √π = π^(1/2) → π^(log_π(16^(1/2)) = 16^(1/2) = 4

Answer 3

Aloha :)

zu a) Aus \(a^{\log(b)}=b^{\log(a)}\) folgt sofort: \(\pi^{\log_\pi\left((x-1)^2\right)}=\left((x-1)^2\right)^{\log_\pi(\pi)}=(x-1)^2\)

zu b) \(\ln(y^4)+4\ln(y^{-1})=4\ln(y)+4\cdot(-1)\ln(y)=4\ln(y)-4\ln(y)=0\)

zu c) Ähnlich zu (a): \(\sqrt\pi^{\log_\pi(16)}=16^{\log_\pi(\sqrt\pi)}=16^{\log_\pi(\pi^{1/2})}=16^{\frac{1}{2}\log_\pi(\pi)}=16^{\frac{1}{2}}=4\)