Proposition: Logarithm to a General Base

The exponential function of general base $\mathbb R\to\mathbb R,~x\to a^x$, is invertible for all positive bases $a > 0$. Its inverse function is continuous, strictly monotonically increasing and called the logarithm to the base $a$ \[\log_a:\mathbb R_{+}^*\to\mathbb R.\]

Furthermore, $\log_a$ (the logarithm to the base $a$) can be calculated using $\ln$ (i.e. the natural logarithm) by the formula $$\log_a(x)=\frac{\ln (x)}{\ln(a)},$$ for all $x\in\mathbb R_{+}^*.$

Proofs: 1

Proofs: 1
Topics: 2

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983