6.1 The functions a^x and e^x — Learn

Prerequisites: the laws of indices for whole-number and fractional powers from Chapter 2; reading a graph and its asymptote from Chapter 2; the idea of a function and its range.

By the end of this section you can: sketch the graph of $y = a^x$ for a positive base $a$; state its intercept, range and asymptote; describe how the shape changes with the base; and sketch the graph of $y = e^x$, explaining what is special about the number $e$.

1. An exponential function

An exponential function has the variable in the index. The simplest examples are

The base $a$ must be positive, because a negative base gives no meaning to powers such as $a^{1/2}$. The base also cannot be $1$, since $1^x = 1$ for every $x$ and the graph would be a flat line.

Every curve $y = a^x$ with $a > 0$ passes through the point $(0, 1)$, because $a^0 = 1$. The $y$-intercept is therefore always $1$. The curve never touches or crosses the $x$-axis: the value is positive for every $x$. The line $y = 0$ is a horizontal asymptote. The domain is all real numbers, and the range is $y > 0$.

The size of the base sets the shape:

• when $a > 1$, the curve rises and gets steeper as $x$ grows;
• when $0 < a < 1$, the curve falls and gets flatter as $x$ grows;
• a larger base rises faster, so $3^x$ is above $2^x$ for $x > 0$.

Example 1. Sketch $y = 4^x$, and state its intercept, asymptote and range.

The curve rises because $4 > 1$. It passes through $(0, 1)$. The asymptote is $y = 0$, and the range is $y > 0$. The graph is steeper than $y = 2^x$, because the base is larger.

Example 2. Describe how the graph of $y = \left(\dfrac{1}{3}\right)^x$ differs from the graph of $y = 3^x$.

The base $\dfrac{1}{3}$ lies between $0$ and $1$, so $y = \left(\dfrac{1}{3}\right)^x$ falls. It is the reflection of $y = 3^x$ in the $y$-axis, because $\left(\dfrac{1}{3}\right)^x = 3^{-x}$. Both curves pass through $(0, 1)$ and both have asymptote $y = 0$.

Example 3. Work out the value of $y = 2^x$ when $x = 3$, $x = 0$ and $x = -2$.

The values rise as $x$ rises, and they stay positive even when $x$ is negative.

Example 4. Write $y = 5^x$ in the form $y = e^{kx}$, giving $k$ exactly.

Write the base as a power of $e$: $5 = e^{\ln 5}$. Then

so $k = \ln 5 \approx 1.609$. Every exponential function can be written with base $e$ in this way.

2. The number e

The number $e$ is a constant, like $\pi$. Its value is

The graph of $y = e^x$ lies between $y = 2^x$ and $y = 3^x$, because $e$ lies between $2$ and $3$. It passes through $(0, 1)$ and rises for every $x$, with asymptote $y = 0$.

The number $e$ is chosen for a special reason. At the point $(0, 1)$ the tangent to $y = e^x$ has gradient exactly $1$. More than that, the gradient of $e^x$ at every point equals its value at that point. No other base has this property. Section 6.2 makes that statement precise, and it is why $e$ is called the natural base for growth and decay.

3. Try it: the exponential explorer

Predict first. The figure shows $y = a^x$ with a slider for the base $a$, and the dashed curve $y = e^x$ for comparison. Before you move the slider, predict the shape when $a = 0.5$, and predict whether the curve rises or falls when $a = 3$. Write both predictions down.

Test. Set $a = 2$ and read the value at $x = 1$. Change $a$ to $3$ and read it again. Then set $a = 0.5$ and read the value at $x = 1$ and at $x = -1$.

Explain. Write two sentences. The first explains why every curve passes through the same point $(0, 1)$, whatever the base. The second explains why the curve falls when $a$ is between $0$ and $1$.

Watch out. These mistakes are common.

Reading $a^x$ as a power of $x$. In $y = x^2$ the variable is the base; in $y = 2^x$ the variable is the exponent. The second curve grows far faster. For $x = 10$, $x^2 = 100$ but $2^{10} = 1024$.

Thinking the curve touches the $x$-axis. The value of $a^x$ becomes very small as $x$ becomes very negative, but it never reaches zero. The $x$-axis is an asymptote, not a crossing point.

Using a negative or zero base. The expression $(-2)^x$ has no real value for $x = \dfrac{1}{2}$, and $0^x$ is $0$ for every positive $x$. So the base must be positive.

Re-check: for your sketch, confirm the $y$-intercept is $(0, 1)$, the curve stays above the $x$-axis, and the direction matches the size of the base.

You should now be able to

• sketch $y = a^x$ for a positive base $a$;
• state the $y$-intercept, the asymptote and the range of $y = a^x$;
• decide whether the curve rises or falls from the size of the base;
• compare two exponential curves with different bases;
• sketch $y = e^x$ and explain what makes the base $e$ special;
• write any $a^x$ in the form $e^{kx}$.