The equation you are thinking of is from the Ebers-Moll BJT model, simplified for the forward-active operating region:
$$I_E = I_{ES}\left(\exp\left(\frac{V_{BE}}{V_T}\right) - 1 \right)$$
Note this gives the emitter current, not the collector current.
The other two equations in the Ebers-Moll model are
$$I_C = \alpha{}I_E$$
$$I_B = (1-\alpha)I_E$$
Unfortunately these equations don't help you find \$\beta\$ because \$\alpha\$ is just \$\beta\$ in disguise:
$$\beta=\frac{I_C}{I_B}=\frac{\alpha}{1-\alpha}$$
Small related question, is the \$I_S\$ ... stated anywhere in a transistor datasheet?
It's usually not in the datasheet, but if a SPICE model is available for your device, you can get a "typical" value from there. Look for the IS parameter in the MODEL card.
Given that \$\alpha\$ and \$\beta\$ are related by \$\alpha = \frac{\beta}{1+\beta}\$ as stated in the wiki article, obviously you can do your sums with either.
However, which is going to be easier to use? I personally always use \$\beta\$, regardless of the transistor configuration.
In common emitter \$I_c = \beta\times I_b\$, so I can say 'I need to control \$I_c\$ collector current, I need at least \$\frac{I_c}{\beta}\$ of base current to do it'.
But as \$\beta >> 1\$ (for most transistors), \$\alpha \approx 1\$, and \$I_c \approx I_e\$. You may object to the approximation, but given the way that \$\beta\$ varies with temperature, \$I_c\$, and between transistors of the same type, that is a far far better approximation than insisting that \$\beta\$ is constant. Any good transistor design will allow for operation with a range of \$\beta\$, at least \$2:1\$, preferably more.
Once you have made the approximation \$I_c \approx I_e\$, then common collector operation is given by 'I need to allow for a base current of \$\frac{I_c}{\beta}\$ to flow in the base circuit, without upsetting operation'.
With a common base stage, you say much the same thing, allowing an amount of base current, however you also say that the emitter to collector gain is slightly less than \$1\$, a fraction of \$\frac{1}{\beta}\$ less than one. The error of the gain from \$1\$ will usually be a smaller error than resistor tolerances and other sources of gain error.
Given that you can write an equation for \$\alpha\$, does that mean that you need to? For most practical engineering designs, the answer is no. If you are in college, and the tutor really likes to use \$\alpha\$, then the answer is yes.
Best Answer
And for what it's worth there are multiple \$\beta\$ factors to consider when designing BJT circuits:
\$\beta\$, \$\beta_{DC}\$, \$h_{FE}\$ - These refer to the BJT's DC forward current gain when the BJT is operating in forward-active mode (small signal amplification). Used when performing the DC design/analysis.
(n.b. The 'FE' refers to Forward current gain, common-Emitter configuration.)
(n.b. Parameters written with UPPERCASE subscripts typically refer to DC parameters.)
\$\beta_{ac}\$, \$h_{fe}\$ - These refer to the BJT's AC forward current gain when the BJT is operating in forward-active mode (small signal amplification). Used when performing the AC design/analysis. Typically, \$\beta_{ac}\ll\beta_{DC}\$.
(n.b. Parameters written with lowercase subscripts typically refer to AC parameters.)
\$\beta_{sat}\$ - Refers to the BJT's forward current gain when the BJT is operating in "hard" saturation mode (when the BJT is turned ON fully). Typical values are \$5 \le \beta_{sat} \le 30\$ with \$\beta_{sat}=I_{C(sat)}/I_{B(sat)}=10\$ being a fairly common value for low power and medium power transistors.
(n.b. Data sheets often use lowercase text for parenthetical text—e.g., \$I_{C(sat)}\$, \$V_{BE(sat)}\$, etc. This does not indicate or imply that these are AC parameters.)
(n.b. \$\beta_{sat}\$ is not an AC parameter because the transistor is not performing small-signal AC amplification when it is operating in saturation mode.)
And as a general rule do not use \$\beta\$, \$\beta_{DC}\$, \$h_{FE}\$ to perform saturation calculations. If you do, the BJT will likely operate in "soft" saturation—i.e., in the transition region between the BJT's forward-active mode and its "hard" saturation (fully ON) mode.