For $y = \log_2 x$, state the domain, range and asymptote, and give four points on the graph.

Domain $x > 0$, range all reals, asymptote $x = 0$; points $\left(\frac{1}{2}, -1\right)$, $(1, 0)$, $(2, 1)$, $(4, 2)$

$f(x) = 3^x$ passes through $(2, 9)$. Write the inverse of $f$ and give the matching point on its graph.

$f^{-1}(x) = \log_3 x$, through $(9, 2)$

Graph $y = \log_3(x - 2) + 1$: give the asymptote, domain, $x$-intercept and three key points.

Asymptote $x = 2$, domain $x > 2$, $x$-intercept $\frac{7}{3}$; points $\left(\frac{7}{3}, 0\right)$, $(3, 1)$, $(5, 2)$

For $y = -2 \log(x + 4)$, find the asymptote, domain, range, both intercepts, and say whether it rises or falls.

Asymptote $x = -4$, domain $x > -4$, range all reals; $x$-intercept $-3$, $y$-intercept $-2 \log 4 \approx -1.204$; it falls

For $f(x) = 2 \log_2(x + 3) - 4$, find the domain, range, asymptote, both intercepts, and the equation of $f^{-1}(x)$.

Domain $x > -3$, range all reals, asymptote $x = -3$; $x$-intercept $1$, $y$-intercept $2 \log_2 3 - 4 \approx -0.830$; $f^{-1}(x) = 2^{(x + 4)/2} - 3$

Grade 12 · MHF4U · Advanced Functions

MHF4U: Graphing Logarithmic Functions

The graph of $y = \log_b x$ is the graph of $y = b^x$ reflected in the line $y = x$ , because the two functions are inverses. Everything about the log graph comes from that: swap the coordinates of the exponential's points and you have the log's.

How It Works

Key points. $y = b^x$ passes through $\left(-1, \frac{1}{b}\right)$ , $(0, 1)$ and $(1, b)$ , so $y = \log_b x$ passes through $\left(\frac{1}{b}, -1\right)$ , $(1, 0)$ and $(b, 1)$ . Those three points and the asymptote are enough to sketch it.

Asymptote, domain and range. The exponential's horizontal asymptote $y = 0$ becomes a vertical asymptote $x = 0$ . You can only take the log of a positive number, so the domain is $x > 0$ ; the range is all real numbers. For $b > 1$ the graph rises slowly to the right and drops toward $-\infty$ as $x$ approaches $0$ .

Transformations. For $y = a \log_b(k(x - d)) + c$ , each point $(x, y)$ of $y = \log_b x$ maps to $\left(\frac{x}{k} + d, ay + c\right)$ . The asymptote moves with the horizontal shift to $x = d$ , and for $k > 0$ the domain is $x > d$ . A vertical shift or stretch leaves the asymptote alone. A negative $a$ reflects the graph in the $x$ -axis, so it falls instead of rising.

Intercepts. For the $x$ -intercept set $y = 0$ and rewrite in exponential form. For the $y$ -intercept put $x = 0$ , which only exists when $0$ is in the domain.

The inverse. To find the inverse of a log function, swap $x$ and $y$ , isolate the log, and rewrite in exponential form. The inverse of a log is an exponential and the other way around.

5 Worked Examples

Easy first, test level last. Try each one on paper, then open the solution.

Easy
For $y = \log_2 x$ , state the domain, range and asymptote, and give four points on the graph.
Show solution
1. Domain $x > 0$ , range all real numbers, vertical asymptote $x = 0$ .
2. $\log_2 \frac{1}{2} = -1$ , $\log_2 1 = 0$ , $\log_2 2 = 1$ , $\log_2 4 = 2$ .
Answer: Domain $x > 0$ , range all reals, asymptote $x = 0$ ; points $\left(\frac{1}{2}, -1\right)$ , $(1, 0)$ , $(2, 1)$ , $(4, 2)$
Easy
$f(x) = 3^x$ passes through $(2, 9)$ . Write the inverse of $f$ and give the matching point on its graph.
Show solution
1. The inverse of $y = 3^x$ is $y = \log_3 x$ .
2. Swap the coordinates: $(9, 2)$ , and indeed $\log_3 9 = 2$ .
Answer: $f^{-1}(x) = \log_3 x$ , through $(9, 2)$
Medium
Graph $y = \log_3(x - 2) + 1$ : give the asymptote, domain, $x$ -intercept and three key points.
Show solution
1. Shift right $2$ and up $1$ . The asymptote moves to $x = 2$ , so the domain is $x > 2$ .
2. Key points of $y = \log_3 x$ are $\left(\frac{1}{3}, -1\right)$ , $(1, 0)$ , $(3, 1)$ . Map $(x, y) \to (x + 2, y + 1)$ : $\left(\frac{7}{3}, 0\right)$ , $(3, 1)$ , $(5, 2)$ .
3. $x$ -intercept: $\log_3(x - 2) = -1$ , so $x - 2 = \frac{1}{3}$ and $x = \frac{7}{3}$ , which is the first key point.
Answer: Asymptote $x = 2$ , domain $x > 2$ , $x$ -intercept $\frac{7}{3}$ ; points $\left(\frac{7}{3}, 0\right)$ , $(3, 1)$ , $(5, 2)$
Medium
For $y = -2 \log(x + 4)$ , find the asymptote, domain, range, both intercepts, and say whether it rises or falls.
Show solution
1. Shift left $4$ : asymptote $x = -4$ , domain $x > -4$ . The range stays all real numbers.
2. $x$ -intercept: $\log(x + 4) = 0$ , so $x + 4 = 1$ and $x = -3$ .
3. $y$ -intercept: $y = -2 \log 4 \approx -1.204$ .
4. $a = -2$ reflects the graph in the $x$ -axis, so it falls from left to right. Check with a point: $(10, 1)$ on $y = \log x$ maps to $(6, -2)$ .
Answer: Asymptote $x = -4$ , domain $x > -4$ , range all reals; $x$ -intercept $-3$ , $y$ -intercept $-2 \log 4 \approx -1.204$ ; it falls
Test level
For $f(x) = 2 \log_2(x + 3) - 4$ , find the domain, range, asymptote, both intercepts, and the equation of $f^{-1}(x)$ .
Show solution
1. Shift left $3$ : asymptote $x = -3$ , domain $x > -3$ , range all real numbers.
2. $x$ -intercept: $2 \log_2(x + 3) = 4$ , so $\log_2(x + 3) = 2$ , $x + 3 = 4$ , $x = 1$ .
3. $y$ -intercept: $f(0) = 2 \log_2 3 - 4 \approx -0.830$ .
4. Inverse: swap, $x = 2 \log_2(y + 3) - 4$ . Then $\frac{x + 4}{2} = \log_2(y + 3)$ , so $y + 3 = 2^{(x + 4)/2}$ .
5. Check: $f(1) = 0$ , and $f^{-1}(0) = 2^2 - 3 = 1$ .
Answer: Domain $x > -3$ , range all reals, asymptote $x = -3$ ; $x$ -intercept $1$ , $y$ -intercept $2 \log_2 3 - 4 \approx -0.830$ ; $f^{-1}(x) = 2^{(x + 4)/2} - 3$

Learn It on Martian Lab

Martian Lab teaches this in the Graphing Exponential & Logarithmic Functions lesson of its MHF4U Exponential & Logarithmic Functions unit. Watch it, then take the Exponential & Logarithmic Functions practice tests to see if it stuck.

Watch the Graphing Exponential & Logarithmic Functions lesson free

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Free Exponential & Logarithmic Functions practice test, with solutions

More Exponential & Logarithmic Functions topics: Laws of Logarithms · Solving Logarithmic Equations · Solving Exponential Equations · Exponential and Log Word Problems

More MHF4U topics: Long and Synthetic Division · Vertical and Horizontal Asymptotes · Proving Trig Identities · Solving Trig Equations · End Behaviour and Degree · Finding the Equation of a Polynomial · Even and Odd Functions · Remainder and Factor Theorem · Solving Polynomial Equations · Polynomial Inequalities · Sum and Difference of Cubes · Holes in Rational Functions · Graphing Rational Functions · Rational Equations and Inequalities · Radian Measure · Unit Circle and Special Angles · Compound Angle Formulas · Double Angle Formulas · Graphing Sine and Cosine with Transformations · Graphs of Tan, Sec, Csc and Cot · Sinusoidal Word Problems · Sums, Products and Quotients of Functions · Composition of Functions · Average vs Instantaneous Rate of Change

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