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]

[581] **Forster Otto**: “Analysis 1, Differential- und Integralrechnung einer VerĂ¤nderlichen”, Vieweg Studium, 1983