Graphs of Exponential and Logarithmic Functions

Exponential graphs

Every graph of $y = b^x$, with $b > 0$ and $b \ne 1$, passes through $(0, 1)$, because $b^0 = 1$, and through $(1, b)$. If $b > 1$ it rises, faster and faster: growth. If $0 < b < 1$ it falls: decay. Either way, the values stay positive and approach $0$ on one side without reaching it, so the $x$-axis is a horizontal asymptote.

• y = 2ˣ
• y = (1/2)ˣ

Since $\left(\tfrac{1}{2}\right)^x = 2^{-x}$, the decay curve is the growth curve reflected in the $y$-axis.

Logarithmic graphs

The logarithm $y = \log_b x$ is the inverse of $y = b^x$, so its graph is the exponential graph reflected in the line $y = x$.

• y = 2ˣ
• y = log₂ x
• y = x

Why the logarithm graph is the exponential flipped

An inverse function swaps inputs and outputs. The point $(2, 4)$ on $y = 2^x$ says $2^2 = 4$; the same fact read backward, $\log_2 4 = 2$, is the point $(4, 2)$ on the logarithm. Swapping the coordinates of every point reflects the graph in $y = x$. A logarithm’s graph is the exponential’s graph with $x$ and $y$ exchanged, so every feature is exchanged too:

	$y = b^x$	$y = \log_b x$
key point	$(0, 1)$	$(1, 0)$
asymptote	horizontal, $y = 0$	vertical, $x = 0$
domain	all real numbers	$x > 0$
range	$y > 0$	all real numbers

Shifts and stretches

The familiar transformations apply. For $y = a \cdot b^{x - h} + k$ and $y = a\log_b(x - h) + k$:

• A shift up by $k$ moves the exponential’s asymptote to $y = k$.
• A shift right by $h$ moves the logarithm’s asymptote to $x = h$ and its domain to $x > h$.
• The factor $a$ stretches vertically, and a negative $a$ reflects the graph in the $x$-axis.

To sketch, draw the asymptote first, then move two or three key points.

Worked examples

Common mistakes

Practice problems