Exponential function

An exponential function is $f(x) = b^x$ for a fixed base $b > 0$, $b \ne 1$. It starts at height 1 at $x = 0$. For a whole number n, $b^n$ multiplies by b n times: $2^3 = 2 \cdot 2 \cdot 2 = 8$. A step to the left divides instead, so $b^{-n} = 1/b^n$ and $b^0 = 1$. Fractions follow from the same rule: two half steps make one whole step, so a half step multiplies by the positive number whose square is b, $b^{1/2} = \sqrt{b}$. In the same way $b^{1/q}$ is the positive q-th root of b and $b^{p/q} = (b^{1/q})^p$. An irrational exponent such as √2 is reached through the rational exponents close to it, which fills in the curve without gaps.

Because $b^{x + \Delta x} = b^x \cdot b^{\Delta x}$, a step of Δx to the right multiplies the height by $b^{\Delta x}$, wherever it starts. The ladder from P shows that factor beside every rise; going left, each step divides by it. A linear function does the opposite: equal steps add the same amount. This one rule fixes the whole shape. The curve passes through (0, 1), and repeated halving never reaches 0, so it stays above the x-axis. For $b > 1$ it rises more and more steeply to the right and creeps toward 0 on the left; for $0 < b < 1$ it is the other way round.

Since $(1/b)^x = b^{-x}$, the graph of $(1/b)^x$ is the graph of $b^x$ reflected in the y-axis: a step right with one is a step left with the other. The base 1 is left out. Every step multiplies by 1, the graph is the line y = 1, and different x give the same value, so there is no inverse. For every other base the inverse function is the logarithm $\log_b x$.

Over a step of length h the curve climbs at the average rate $\dfrac{b^{x + h} - b^x}{h} = b^x \cdot \dfrac{b^h - 1}{h}$. The second factor does not depend on x, and as h shrinks to 0 it approaches a number that depends only on b, the natural logarithm $\ln b$. So the slope of the tangent is $(b^x)' = \ln b \cdot b^x$: the higher the curve, the steeper it is, always in the same proportion. The base for which that proportion is 1 is $e \approx 2.718$. The graph of $e^x$ has a slope equal to its height everywhere, because $\ln e = 1$.