Question

logarithmics

Answer 1

\[log_b(a) = k \iff b^k = a\]

Answer 2

\[\log_{x} y=64, find \log_{x^{2}}y ^{3} \]

Answer 3

Well, using what I said originally..

\[log_xy = 64 \implies x^{64} = y\]
\[\text{Let }k= log_{x^2}y^3 = 3(log_{x^2}y) \implies \frac{k}{3} = log_{x^2}y\]\[\implies (x^2)^{\frac{k}{3} } = y = x^{64}\]\[\implies x^{\frac{2k}{3}}= x^{64}\]\[\implies \frac{2k}{3} = 64 = 32(2)(\frac{3}{3})\]\[\implies k = 32\times 3 = 96\]\[\implies log_{x^2}y^3 = 96.\]