Graphs of Logarithmic Functions

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

1. 1. ⓐ
   2. ⓑ
   3. ⓒ

      Find the domain of the function \(f(x)=\frac{5}{x-2}\)

   4. ⓓ

      Find the domain of the function \(f(x)={\log }_{2}(x-5)\) .

   ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

   1. ⓐ

      The set of points on the graph is \(\{(-4,-2),(-2,-1),(-1,1),(1,2)\}\)
      The Domain is the set of all x-coordinates: \(\{-4,-2,-1,1\}\)
      The Range is the set of all y-coordinates: \(\{-2,-1,1\}\)
      Notice that even though y-coodinate of 1 appears twice, we only list it once.

   2. ⓑ

      Domain: \((-\infty ,\infty )\)
      Range: \([-2,\infty )\)
      Notice that \(-2\) is included because the point \((3,-2)\) is on the graph of a function.

   3. ⓒ

      A function is not defined when the denominator is zero. We need to set the denominator equal zero and exclude this value(s) from the domain.
      \(x-2=0,x=2,\) Domain \((-\infty ,2)\cup (2,\infty )\)
      Notice that 2 is excluded from the domain because the function is not defined at \(x=2\)

   4. ⓓ

      From the definition of the logarithmic function \(f(x)={\log }_{a}x\) we know that \(x>0\)
      To find domain of \(f(x)={\log }_{2}(x-5\) , we need to set up and solve inequality.
      \(x-5>0\) ,
      \(x>5)\) Domain: \((5,\infty )\)

2. Find the domain and range of a relation.

3. Find the domain and the range of the function graphed. Use interval notation.

4. Find the domain of the function \(f(x)={\log }_{2}(x+4)\) . Notice: this is the same function that was graphed in question 2.

5. Graph \(y={\text{log}}_{2}x.\)

   ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

   To graph the function, we will first rewrite the logarithmic equation, \(y={\text{log}}_{2}x,\) in exponential form, \({2}^{y}=x.\)

   We will use point plotting to graph the function. It will be easier to start with values of y and then get x.

   \(y\)	\({2}^{y}=x\)	\((x,y)\)
   \(-2\)	\({2}^{-2}=\frac{1}{{2}^{2}}=\frac{1}{4}\)	\((\frac{1}{4},2)\)
   \(-1\)	\({2}^{-1}=\frac{1}{{2}^{1}}=\frac{1}{2}\)	\((\frac{1}{2},-1)\)
   0	\({2}^{0}=1\)	\((1,0)\)
   1	\({2}^{1}=2\)	\((2,1)\)
   2	\({2}^{2}=4\)	\((4,2)\)
   3	\({2}^{3}=8\)	\((8,3)\)

6. Graph \(y={\text{log}}_{3}x\) and \(y={\text{log}}_{5}x\) in the same coordinate system.
   

   \(y\)	\({3}^{y}=x\)	\((x,y)\)
   \(y\)	\({5}^{y}=x\)	\((x,y)\)

7. Graph \(y={\log }_{1/3}x\)

   \(y\)	\({(\frac{1}{3})}^{y}=x\)	\((x,y)\)

8. 
   Do the graphs of \(y={\log }_{2}x\) , \(y={\log }_{3}x\) , and \(y={\log }_{5}x\) have the shape we expect from a logarithmic function where \(a>0\) ? (Remember a is the base of the log function)

9. Is there a point they all share? Why does this make sense?

10. Do they all have a point \((a,1)\) ? Why does this make sense?

11. Do they all have a point \((\frac{1}{a},-1)\) ? Why does this make sense?

12. Do they all have the same vertical asymptote? What is the equation of the vertical asymptote?

13. Do they all have the same domain? Write the domain in the interval notation.

14. Do they all have the same range? Write the range in the interval notation.

15. What is the domain of \(f(x)={\log }_{2}(x+3)?\)

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    The logarithmic function is defined only when the input is positive, so this function is defined when \(x+3>0.\) Solving this inequality,

    \[\begin{array}{lllll}x+3>0 & \text{The input must be positive}. \\ x>-3\begin{array}{llll} & & & \end{array} & \text{Subtract 3}.\end{array}\]

    The domain of \(f(x)={\log }_{2}(x+3)\) is \((-3,\infty ).\)

16. What is the domain of \(f(x)={\log }_{5}(x-2)+1?\)

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    \((2,\infty )\)

17. What is the domain of \(f(x)=\log (5-2x)?\)

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    The logarithmic function is defined only when the input is positive, so this function is defined when \(5-2x>0.\) Solving this inequality,

    \[\begin{array}{lllll}5-2x>0 & \text{The input must be positive}. \\ -2x>-5 & \text{Subtract }5. \\ x<\frac{5}{2}\begin{array}{llll} & & & \end{array} & \text{Divide by }-2\ \text{and switch the inequality}.\end{array}\]

    The domain of \(f(x)=\log (5-2x)\) is \((-\infty ,\frac{5}{2}).\)

18. What is the domain of \(f(x)=\log (x-5)+2?\)

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    \((5,\infty )\)

19. Graph \(f(x)={\log }_{5}(x).\) State the domain, range, and asymptote.

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    Before graphing, identify the behavior and key points for the graph.

    • Since \(b=5\) is greater than one, we know the function is increasing. The left tail of the graph will approach the vertical asymptote \(x=0,\) and the right tail will increase slowly without bound.
    • The x-intercept is \((1,0).\)
    • The key point \((5,1)\) is on the graph.
    • We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points (see ).

    The domain is \((0,\infty ),\) the range is \((-\infty ,\infty ),\) and the vertical asymptote is \(x=0.\)

20. Graph \(f(x)={\log }_{\frac{1}{5}}(x).\) State the domain, range, and asymptote.

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    The domain is \((0,\infty ),\) the range is \((-\infty ,\infty ),\) and the vertical asymptote is \(x=0.\)

21. Sketch the horizontal shift \(f(x)={\log }_{3}(x-2)\) alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    Since the function is \(f(x)={\log }_{3}(x-2),\) we notice \(x+(-2)=x-2.\)

    Thus \(c=-2,\) so \(c<0.\) This means we will shift the function \(f(x)={\log }_{3}(x)\) right 2 units.

    The vertical asymptote is \(x=-(-2)\) or \(x=2.\)

    Consider the three key points from the parent function, \((\frac{1}{3},-1),\) \((1,0),\) and \((3,1).\)

    The new coordinates are found by adding 2 to the \(x\) coordinates.

    Label the points \((\frac{7}{3},-1),\) \((3,0),\) and \((5,1).\)

    The domain is \((2,\infty ),\) the range is \((-\infty ,\infty ),\) and the vertical asymptote is \(x=2.\)

22. Sketch a graph of \(f(x)={\log }_{3}(x+4)\) alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    The domain is \((-4,\infty ),\) the range \((-\infty ,\infty ),\) and the asymptote \(x=-4.\)

23. Sketch a graph of \(f(x)={\log }_{3}(x)-2\) alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    Since the function is \(f(x)={\log }_{3}(x)-2,\) we will notice \(d=-2.\) Thus \(d<0.\)

    This means we will shift the function \(f(x)={\log }_{3}(x)\) down 2 units.

    The vertical asymptote is \(x=0.\)

    Consider the three key points from the parent function, \((\frac{1}{3},-1),\) \((1,0),\) and \((3,1).\)

    The new coordinates are found by subtracting 2 from the y coordinates.

    Label the points \((\frac{1}{3},-3),\) \((1,-2),\) and \((3,-1).\)

    The domain is \((0,\infty ),\) the range is \((-\infty ,\infty ),\) and the vertical asymptote is \(x=0.\)

    The domain is \((0,\infty ),\) the range is \((-\infty ,\infty ),\) and the vertical asymptote is \(x=0.\)

24. Sketch a graph of \(f(x)={\log }_{2}(x)+2\) alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    The domain is \((0,\infty ),\) the range is \((-\infty ,\infty ),\) and the vertical asymptote is \(x=0.\)

25. Sketch a graph of \(f(x)=2{\log }_{4}(x)\) alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    Since the function is \(f(x)=2{\log }_{4}(x),\) we will notice \(a=2.\)

    This means we will stretch the function \(f(x)={\log }_{4}(x)\) by a factor of 2.

    The vertical asymptote is \(x=0.\)

    Consider the three key points from the parent function, \((\frac{1}{4},-1),\) \((1,0),\) and \((4,1).\)

    The new coordinates are found by multiplying the \(y\) coordinates by 2.

    Label the points \((\frac{1}{4},-2),\) \((1,0)\ ,\) and \((4,\text{2}).\)

    The domain is \((0,\ \infty ),\) the range is \((-\infty ,\infty ),\) and the vertical asymptote is \(x=0.\) See .

    The domain is \((0,\infty ),\) the range is \((-\infty ,\infty ),\) and the vertical asymptote is \(x=0.\)

26. Sketch a graph of \(f(x)=\frac{1}{2}\ {\log }_{4}(x)\) alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    The domain is \((0,\infty ),\) the range is \((-\infty ,\infty ),\) and the vertical asymptote is \(x=0.\)

27. Sketch a graph of \(f(x)=5\log (x+2).\) State the domain, range, and asymptote.

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    Remember: what happens inside parentheses happens first. First, we move the graph left 2 units, then stretch the function vertically by a factor of 5, as in . The vertical asymptote will be shifted to \(x=-2.\) The x-intercept will be \((-1,0).\) The domain will be \((-2,\infty ).\) Two points will help give the shape of the graph: \((-1,0)\) and \((8,5).\) We chose \(x=8\) as the x-coordinate of one point to graph because when \(x=8,\) \(x+2=10,\) the base of the common logarithm.

    The domain is \((-2,\infty ),\) the range is \((-\infty ,\infty ),\) and the vertical asymptote is \(x=-2.\)

28. Sketch a graph of the function \(f(x)=3\log (x-2)+1.\) State the domain, range, and asymptote.

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    The domain is \((2,\infty ),\) the range is \((-\infty ,\infty ),\) and the vertical asymptote is \(x=2.\)

29. Sketch a graph of \(f(x)=\log (-x)\) alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    Before graphing \(f(x)=\log (-x),\) identify the behavior and key points for the graph.

    • Since \(b=10\) is greater than one, we know that the parent function is increasing. Since the input value is multiplied by \(-1,\) \(f\) is a reflection of the parent graph about the y-axis. Thus, \(f(x)=\log (-x)\) will be decreasing as \(x\) moves from negative infinity to zero, and the right tail of the graph will approach the vertical asymptote \(x=0.\)
    • The x-intercept is \((-1,0).\)
    • We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.

    The domain is \((-\infty ,0),\) the range is \((-\infty ,\infty ),\) and the vertical asymptote is \(x=0.\)

30. Graph \(f(x)=-\log (-x).\) State the domain, range, and asymptote.

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    The domain is \((-\infty ,0),\) the range is \((-\infty ,\infty ),\) and the vertical asymptote is \(x=0.\)

31. Solve \(4\ln (x)+1=-2\ln (x-1)\) graphically. Round to the nearest thousandth.

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    Press [Y=] and enter \(4\ln (x)+1\) next to Y₁=. Then enter \(-2\ln (x-1)\) next to Y₂=. For a window, use the values 0 to 5 for \(x\) and –10 to 10 for \(y.\) Press [GRAPH]. The graphs should intersect somewhere a little to right of \(x=1.\)

    For a better approximation, press [2ND] then [CALC]. Select [5: intersect] and press [ENTER] three times. The x-coordinate of the point of intersection is displayed as 1.3385297. (Your answer may be different if you use a different window or use a different value for Guess?) So, to the nearest thousandth, \(x\approx 1.339.\)

32. Solve \(5\log (x+2)=4-\log (x)\) graphically. Round to the nearest thousandth.

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    \(x\approx 3.049\)

33. What is the vertical asymptote of \(f(x)=-2{\log }_{3}(x+4)+5?\)

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    The vertical asymptote is at \(x=-4.\)

34. What is the vertical asymptote of \(f(x)=3+\ln (x-1)?\)

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    \(x=1\)

35. Find a possible equation for the common logarithmic function graphed in .

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    This graph has a vertical asymptote at \(x=-2\) and has been vertically reflected. We do not know yet the vertical shift or the vertical stretch. We know so far that the equation will have form:

    \[f(x)=-a\log (x+2)+k\]

    It appears the graph passes through the points \((-1,1)\) and \((2,-1).\) Substituting \((-1,1),\)

    \[\begin{array}{ll}1=-a\log (-1+2)+k\ \ \ \ \ \ & \text{Substitute }(-1,1). \\ 1=-a\log (1)+k & \text{Arithmetic}. \\ 1=k & \text{log(1)}=0.\end{array}\]

    Next, substituting in \((2,-1)\) ,

    \[\begin{array}{lll}-1=-a\log (2+2)+1 & & \text{Plug in }(2,-1). \\ -2=-a\log (4) & & \text{Arithmetic}. \\ \ a=\frac{2}{\log (4)} & & \text{Solve for }a.\end{array}\]

    This gives us the equation \(f(x)=-\frac{2}{\log (4)}\log (x+2)+1.\)

36. Give the equation of the natural logarithm graphed in .

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    \(f(x)=2\ln (x+3)-1\)

37. The inverse of every logarithmic function is an exponential function and vice-versa. What does this tell us about the relationship between the coordinates of the points on the graphs of each?

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    Since the functions are inverses, their graphs are mirror images about the line \(y=x.\) So for every point \((a,b)\) on the graph of a logarithmic function, there is a corresponding point \((b,a)\) on the graph of its inverse exponential function.

38. What type(s) of translation(s), if any, affect the range of a logarithmic function?

39. What type(s) of translation(s), if any, affect the domain of a logarithmic function?

    ଉତ୍ତରକୁ ଖୋଲନ୍ତୁ

    Shifting the function right or left and reflecting the function about the y-axis will affect its domain.

40. Consider the general logarithmic function \(f(x)={\log }_{b}(x).\) Why can’t \(x\) be zero?

ନିଜେ ଚେଷ୍ଟାକରନ୍ତୁ

Parts of this page are adapted from OpenStax Precalculus 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

ଅଧିକ Precalculus

Complex numbersPolynomial functionsRational functionsSequences and seriesThe binomial theoremConic sectionsVectorsExponential and logarithmic functionsPolynomial division and the remainder theoremParametric equations and polar coordinates