What is the meaning of $\exp(\,\cdot\,)$?

What is the meaning of the notation $\exp(\text{expression})$ ?

I think that it's something of the form $a^\text{expression}$ but does it mean that the base $a=e$ or can it be any base?

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Note that googling "exp" already returns a useful result. –  Pedro Tamaroff Apr 27 '12 at 0:17

$\exp(x)=e^x$

Where $e$ is Euler's Number (approx. $2.718281828$).

This notation is common when $x$ is complicated and messy when put in the form $e^x$ (like large fractions or polynomials, for instance). Also, $\exp$ is the name of the $e^x$ function in many programming languages.

$e$ is a very important constant that appears in many fields of mathematics. The inverse of the $\exp(x)$ function is the natural logarithm (often written as $\log(x)$, $\ln(x)$ or $\log_e(x)$).

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Also note that in Rudin's prologue on the exponential function in his Real And Complex Analysis, he introduces $\exp$ by its power series and then shows that $\exp(a+b) = \exp(a) + \exp(b)$ justifying that it makes sense to write $e^{z}$ instead of $\exp(z)$ (where $e = \exp(1)$). $$\phantom{x}$$Thus it's possible to argue that to introduce $\sin$, $\cos$ etc. one needs power series and a proof that differentiation thereof is done term by term - clearly far beyond the reach of a typical high school (equivalent) class. –  kahen Apr 27 '12 at 13:41

The notation $\exp(x)$ means $e^x$. See here on Wikipedia for example.

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To answer your question, no - it isn't defined for an arbitrary base $a$, only $e$ which is the value you hit when you try to take the limit as $n$ goes to $0$ to the following expression (as Mr. Bernoulli historically encountered the elusive $e$):

$$\lim_{n\rightarrow0}\left(1+\frac{1}{n} \right)^n = L$$

Where $L = e$ or $2.71828182845\ldots$ Finally:

$$\exp(x)=e^x$$

You can see why it would be prudent to use the left side when you have a hefty exponent which is quite natural for the uses of $e$ (think Gaussian function, normal distribution, etc).

Usually, for other bases, especially in computing/programming, you'd use a more generic function such as $\operatorname{pow}(a, b)$ which basically returns $a^b$.

Hope it helps.

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$\exp(x)$ and $e^x$ refer to the same function, but I'm not sure I would say $\exp(x)$ means $e^x$.

Consider the expression $b^x$, where $b$ is real and $x$ is irrational. How would you define it?

I would say, $b^x = \exp(\ln(b^x)) = \exp(x \ln(b))$, where $\exp$ means the antilogarithm. The expression $e^x$ is, of course, a special case of $b^x$. If your book is taking this approach, you might find the theorem $\exp(x) = e^x$ proved later on.

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