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Proof: If <math>p</math> is a positive integer, this just follows from repeated application of the above-mentioned additive property of the logarithm. For <math>p = \frac{1}{n}</math>, note that the statement follows by observing that <math>\ln(x) = \ln((x^{1/n})^n) = n\ln(x^{1/n}</math>. Thus, <math>p\ln(x) = \ln(x^p)</math> for all rational numbers <math>p</math>. The statement must therefore hold for all reals <math>p</math> by continuity.

Proof: If <math>p</math> is a positive integer, this just follows from repeated application of the above-mentioned additive property of the logarithm. For <math>p = \frac{1}{n}</math>, note that the statement follows by observing that <math>\ln(x) = \ln((x^{1/n})^n) = n\ln(x^{1/n})</math>. Thus, <math>p\ln(x) = \ln(x^p)</math> for all rational numbers <math>p</math>. The statement must therefore hold for all reals <math>p</math> by continuity.