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## 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 $x\in\mathbb R_{+}^*.$

| | | | | created: 2017-05-01 18:06:01 | modified: 2017-06-05 06:00:30 | by: bookofproofs | references: [581]

## 1.Proof: (related to "Logarithm to a General Base")

### Bibliography (further reading)

[581] Forster Otto: “Analysis 1, Differential- und Integralrechnung einer Veränderlichen”, Vieweg Studium, 1983