For $a0,\ a\ne1,\ M0$ and real $r$: $\log_a(M^r)=r\log_a M$ Examples ExpressionSi

Question

For $a>0,\ a\ne1,\ M>0$ and real $r$: $\log_a(M^r)=r\log_a M$ Examples ExpressionSimplified form $\log_2(5^x)$ $\log_3\sqrt{2}$ $\lg\sqrt[3]{x},\ x>0$ Watch the brackets For $x>0$: $\lg(2x^3)=\lg2+3\lg x$, whereas $\lg((2x)^3)=3\lg(2x)$. $(\log_a x)^3$ is the cube of the logarithm; the power law does not apply.

Accepted Answer

$\log_a(b^r)=r\log_a b$

Expression	Simplified form
\(\log_2(5^x)\)
\(\log_3\sqrt{2}\)
\(\lg\sqrt[3]{x},\ x>0\)

Power Law and Its Proof

Power Law and Its Proof

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