Logarithm of Power/Natural Logarithm/Proof 3
Jump to navigation
Jump to search
Theorem
Let $x \in \R$ be a strictly positive real number.
Let $a \in \R$ be a real number such that $a > 1$.
Let $r \in \R$ be any real number.
Let $\ln x$ be the natural logarithm of $x$.
Then:
- $\map \ln {x^r} = r \ln x$
Proof
Here we adopt the definition of $\ln x$ to be:
- $\ds \ln x := \int_1^x \dfrac {\d t} t$
\(\ds \map \ln {x^r}\) | \(=\) | \(\ds \int_1^{x^r} \dfrac {\d t} t\) | Definition of Natural Logarithm | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_1^x \dfrac {r t^{r - 1} \rd t} {t^r}\) | Integration by Substitution: $t \mapsto t^r$, $\d t \mapsto r t^{r - 1} \rd t$, $1 \mapsto 1$, $x^r \mapsto x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds r \int_1^x \dfrac {\d t} t\) | Primitive of Constant Multiple of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds r \ln x\) | Definition of Natural Logarithm |
$\blacksquare$