Graphs of Exponential and Logarithmic Functions

Definition Let $b > 0$ with $b \ne 1$. The graph of $y = b^x$ has these features.

• The domain is $\mathbb{R}$ and the range is $(0, \infty)$, so the graph lies entirely above the $x$-axis and has no $x$-intercept.
• It passes through $\left(-1, \frac{1}{b}\right)$, $(0, 1)$ and $(1, b)$.
• If $b > 1$ the function is increasing, with $b^x \to 0$ as $x \to -\infty$ and $b^x$ growing without bound as $x \to \infty$.
• If $0 < b < 1$ the function is decreasing, with $b^x \to 0$ as $x \to \infty$ and $b^x$ growing without bound as $x \to -\infty$.
• The line $y = 0$, the $x$-axis, is a horizontal asymptote.

Definition Let $b > 0$ with $b \ne 1$. The graph of $y = \log_b x$ has these features.

• The domain is $(0, \infty)$ and the range is $\mathbb{R}$, so the graph lies entirely to the right of the $y$-axis and has no $y$-intercept.
• It passes through $\left(\frac{1}{b}, -1\right)$, $(1, 0)$ and $(b, 1)$.
• If $b > 1$ the function is increasing. Its values decrease without bound as $x$ approaches $0$ from the right, and increase without bound, but very slowly, as $x \to \infty$.
• If $0 < b < 1$ the function is decreasing. Its values increase without bound as $x$ approaches $0$ from the right, and decrease without bound as $x \to \infty$.
• The line $x = 0$, the $y$-axis, is a vertical asymptote.

Definition For $b > 0$ with $b \ne 1$, a point $(a, c)$ lies on the graph of $y = b^x$ exactly when $(c, a)$ lies on the graph of $y = \log_b x$, because

Swapping the coordinates of every point is the reflection in the line $y = x$. So the two graphs are mirror images in that line, the horizontal asymptote $y = 0$ of the exponential becomes the vertical asymptote $x = 0$ of the logarithm, and the domain and range trade places.

Definition For any function $f$, the graph of $y = -f(x)$ is the reflection of the graph of $y = f(x)$ in the $x$-axis, which sends each point $(a, c)$ to $(a, -c)$. The graph of $y = f(-x)$ is the reflection in the $y$-axis, which sends $(a, c)$ to $(-a, c)$. For $b > 0$ with $b \ne 1$, two of these reflections give functions you already know,

The second identity holds because $\left(\frac{1}{b}\right)^y = b^{-y}$, so $\left(\frac{1}{b}\right)^y = x$ exactly when $-y = \log_b x$.

Definition For real numbers $h$ and $k$, the graph of $y = f(x - h) + k$ is the graph of $y = f(x)$ shifted $h$ units to the right and $k$ units up, where a negative $h$ means a shift to the left and a negative $k$ a shift down. Each point $(a, c)$ moves to $(a + h, c + k)$, and each asymptote moves with the graph. In particular, for $b > 0$ with $b \ne 1$,

Definition Let $b > 0$ with $b \ne 1$, and let $a \ne 0$, $h$ and $k$ be real constants.

• The function $y = a \cdot b^{x - h} + k$ has domain $\mathbb{R}$ and horizontal asymptote $y = k$. Its range is $(k, \infty)$ when $a > 0$ and $(-\infty, k)$ when $a < 0$.
• The function $y = a \log_b(x - h) + k$ has domain $(h, \infty)$, vertical asymptote $x = h$ and range $\mathbb{R}$.

More generally, the domain of $\log_b g(x)$ is the set of $x$ in the domain of $g$ with $g(x) > 0$, which is how a reflection in the $y$-axis moves a logarithm’s domain to the left of its asymptote.