Topic 8 Exponential functions

If \(f(x)=b^x\) for some \(b\), then \(f(x)\) is called an exponential function.

The notation for the most often encountered exponential function is \(\exp(x):= e^x\),     where \(e=2.7128\) is the Euler’s number.

Graph of the natural logarithm function

Figure 8.1: Graph of the natural logarithm function

Example 8.1 In a circuit containing a capacitor, the instantaneous voltage across the capacitor is given by the equation \[\nu = V\left(1-e^{\frac{-t}{RC}}\right),\] where \(V\) is the initial supply voltage, \(R\) is the resistance, \(C\) is the capacitance and \(t\) is the time since the initial connection of the supply voltage. If \(V=200\,\mathrm{V}\), \(R=10\,\mathrm{k}\Omega\) and \(C=20\,\mu\mathrm{F}\), calculate the time when the voltage reaches \(\nu =100\,\mathrm{V}\).

Solution. We want to make \(t\) the subject of the formula above: \[\begin{align*} \frac{\nu}{V} &=1-e^{\frac{-t}{RC}}\\ e^{\frac{-t}{RC}} &= 1-\frac{\nu}{V}\\ -\frac{t}{RC} &= \ln\left(1-\frac{\nu}{V}\right)\\ t &= -RC\ln\left(1-\frac{\nu}{V}\right)\\ \end{align*}\] Hence \[\begin{align*} t&=-10\,\mathrm{k}\Omega\cdot 20\,\mu\mathrm{F}\cdot \ln\left(1-\frac{100\,\mathrm{V}}{200\,\mathrm{V}}\right)\\ &\approx 138.6\cdot 10^3\cdot 10^{-6} \,\Omega\mathrm{F}\\ &= 138.6 \,\mathrm{ms} \end{align*}\] (since \(\Omega\mathrm{F} = \mathrm{s}\)).