# 4.4 Graphs of logarithmic functions  (Page 8/8)

 Page 8 / 8

## Verbal

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?

Since the functions are inverses, their graphs are mirror images about the line $\text{\hspace{0.17em}}y=x.\text{\hspace{0.17em}}$ So for every point $\text{\hspace{0.17em}}\left(a,b\right)\text{\hspace{0.17em}}$ on the graph of a logarithmic function, there is a corresponding point $\text{\hspace{0.17em}}\left(b,a\right)\text{\hspace{0.17em}}$ on the graph of its inverse exponential function.

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

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

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

Consider the general logarithmic function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right).\text{\hspace{0.17em}}$ Why can’t $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ be zero?

Does the graph of a general logarithmic function have a horizontal asymptote? Explain.

No. A horizontal asymptote would suggest a limit on the range, and the range of any logarithmic function in general form is all real numbers.

## Algebraic

For the following exercises, state the domain and range of the function.

$f\left(x\right)={\mathrm{log}}_{3}\left(x+4\right)$

$h\left(x\right)=\mathrm{ln}\left(\frac{1}{2}-x\right)$

Domain: $\text{\hspace{0.17em}}\left(-\infty ,\frac{1}{2}\right);\text{\hspace{0.17em}}$ Range: $\text{\hspace{0.17em}}\left(-\infty ,\infty \right)$

$g\left(x\right)={\mathrm{log}}_{5}\left(2x+9\right)-2$

$h\left(x\right)=\mathrm{ln}\left(4x+17\right)-5$

Domain: $\text{\hspace{0.17em}}\left(-\frac{17}{4},\infty \right);\text{\hspace{0.17em}}$ Range: $\text{\hspace{0.17em}}\left(-\infty ,\infty \right)$

$f\left(x\right)={\mathrm{log}}_{2}\left(12-3x\right)-3$

For the following exercises, state the domain and the vertical asymptote of the function.

$\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x-5\right)$

Domain: $\text{\hspace{0.17em}}\left(5,\infty \right);\text{\hspace{0.17em}}$ Vertical asymptote: $\text{\hspace{0.17em}}x=5$

$\text{\hspace{0.17em}}g\left(x\right)=\mathrm{ln}\left(3-x\right)$

$\text{\hspace{0.17em}}f\left(x\right)=\mathrm{log}\left(3x+1\right)$

Domain: $\text{\hspace{0.17em}}\left(-\frac{1}{3},\infty \right);\text{\hspace{0.17em}}$ Vertical asymptote: $\text{\hspace{0.17em}}x=-\frac{1}{3}$

$\text{\hspace{0.17em}}f\left(x\right)=3\mathrm{log}\left(-x\right)+2$

$\text{\hspace{0.17em}}g\left(x\right)=-\mathrm{ln}\left(3x+9\right)-7$

Domain: $\text{\hspace{0.17em}}\left(-3,\infty \right);\text{\hspace{0.17em}}$ Vertical asymptote: $\text{\hspace{0.17em}}x=-3$

For the following exercises, state the domain, vertical asymptote, and end behavior of the function.

$f\left(x\right)=\mathrm{ln}\left(2-x\right)$

$f\left(x\right)=\mathrm{log}\left(x-\frac{3}{7}\right)$

Domain: $\left(\frac{3}{7},\infty \right)$ ;
Vertical asymptote: $x=\frac{3}{7}$ ; End behavior: as $x\to {\left(\frac{3}{7}\right)}^{+},f\left(x\right)\to -\infty$ and as $x\to \infty ,f\left(x\right)\to \infty$

$h\left(x\right)=-\mathrm{log}\left(3x-4\right)+3$

$g\left(x\right)=\mathrm{ln}\left(2x+6\right)-5$

Domain: $\left(-3,\infty \right)$ ; Vertical asymptote: $x=-3$ ;
End behavior: as $x\to -{3}^{+}$ , $f\left(x\right)\to -\infty$ and as $x\to \infty$ , $f\left(x\right)\to \infty$

$f\left(x\right)={\mathrm{log}}_{3}\left(15-5x\right)+6$

For the following exercises, state the domain, range, and x - and y -intercepts, if they exist. If they do not exist, write DNE.

$h\left(x\right)={\mathrm{log}}_{4}\left(x-1\right)+1$

Domain: $\text{\hspace{0.17em}}\left(1,\infty \right);\text{\hspace{0.17em}}$ Range: $\text{\hspace{0.17em}}\left(-\infty ,\infty \right);\text{\hspace{0.17em}}$ Vertical asymptote: $\text{\hspace{0.17em}}x=1;\text{\hspace{0.17em}}$ x -intercept: $\text{\hspace{0.17em}}\left(\frac{5}{4},0\right);\text{\hspace{0.17em}}$ y -intercept: DNE

$f\left(x\right)=\mathrm{log}\left(5x+10\right)+3$

$g\left(x\right)=\mathrm{ln}\left(-x\right)-2$

Domain: $\text{\hspace{0.17em}}\left(-\infty ,0\right);\text{\hspace{0.17em}}$ Range: $\text{\hspace{0.17em}}\left(-\infty ,\infty \right);\text{\hspace{0.17em}}$ Vertical asymptote: $\text{\hspace{0.17em}}x=0;\text{\hspace{0.17em}}$ x -intercept: $\text{\hspace{0.17em}}\left(-{e}^{2},0\right);\text{\hspace{0.17em}}$ y -intercept: DNE

$f\left(x\right)={\mathrm{log}}_{2}\left(x+2\right)-5$

$h\left(x\right)=3\mathrm{ln}\left(x\right)-9$

Domain: $\text{\hspace{0.17em}}\left(0,\infty \right);\text{\hspace{0.17em}}$ Range: $\text{\hspace{0.17em}}\left(-\infty ,\infty \right);\text{\hspace{0.17em}}$ Vertical asymptote: $\text{\hspace{0.17em}}x=0;\text{\hspace{0.17em}}$ x -intercept: $\text{\hspace{0.17em}}\left({e}^{3},0\right);\text{\hspace{0.17em}}$ y -intercept: DNE

## Graphical

For the following exercises, match each function in [link] with the letter corresponding to its graph.

$d\left(x\right)=\mathrm{log}\left(x\right)$

$f\left(x\right)=\mathrm{ln}\left(x\right)$

B

$g\left(x\right)={\mathrm{log}}_{2}\left(x\right)$

$h\left(x\right)={\mathrm{log}}_{5}\left(x\right)$

C

$j\left(x\right)={\mathrm{log}}_{25}\left(x\right)$

For the following exercises, match each function in [link] with the letter corresponding to its graph.

$f\left(x\right)={\mathrm{log}}_{\frac{1}{3}}\left(x\right)$

B

$g\left(x\right)={\mathrm{log}}_{2}\left(x\right)$

$h\left(x\right)={\mathrm{log}}_{\frac{3}{4}}\left(x\right)$

C

For the following exercises, sketch the graphs of each pair of functions on the same axis.

$f\left(x\right)=\mathrm{log}\left(x\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)={10}^{x}$

$f\left(x\right)=\mathrm{log}\left(x\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)={\mathrm{log}}_{\frac{1}{2}}\left(x\right)$

$f\left(x\right)={\mathrm{log}}_{4}\left(x\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=\mathrm{ln}\left(x\right)$

$f\left(x\right)={e}^{x}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=\mathrm{ln}\left(x\right)$

For the following exercises, match each function in [link] with the letter corresponding to its graph.

$f\left(x\right)={\mathrm{log}}_{4}\left(-x+2\right)$

$g\left(x\right)=-{\mathrm{log}}_{4}\left(x+2\right)$

C

$h\left(x\right)={\mathrm{log}}_{4}\left(x+2\right)$

For the following exercises, sketch the graph of the indicated function.

$\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{2}\left(x+2\right)$

$\text{\hspace{0.17em}}f\left(x\right)=2\mathrm{log}\left(x\right)$

$\text{\hspace{0.17em}}f\left(x\right)=\mathrm{ln}\left(-x\right)$

$g\left(x\right)=\mathrm{log}\left(4x+16\right)+4$

$g\left(x\right)=\mathrm{log}\left(6-3x\right)+1$

$h\left(x\right)=-\frac{1}{2}\mathrm{ln}\left(x+1\right)-3$

For the following exercises, write a logarithmic equation corresponding to the graph shown.

Use $\text{\hspace{0.17em}}y={\mathrm{log}}_{2}\left(x\right)\text{\hspace{0.17em}}$ as the parent function.

$\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{2}\left(-\left(x-1\right)\right)$

Use $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{3}\left(x\right)\text{\hspace{0.17em}}$ as the parent function.

Use $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{4}\left(x\right)\text{\hspace{0.17em}}$ as the parent function.

$f\left(x\right)=3{\mathrm{log}}_{4}\left(x+2\right)$

Use $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{5}\left(x\right)\text{\hspace{0.17em}}$ as the parent function.

## Technology

For the following exercises, use a graphing calculator to find approximate solutions to each equation.

$\mathrm{log}\left(x-1\right)+2=\mathrm{ln}\left(x-1\right)+2$

$x=2$

$\mathrm{log}\left(2x-3\right)+2=-\mathrm{log}\left(2x-3\right)+5$

$\mathrm{ln}\left(x-2\right)=-\mathrm{ln}\left(x+1\right)$

$x\approx \text{2}\text{.303}$

$2\mathrm{ln}\left(5x+1\right)=\frac{1}{2}\mathrm{ln}\left(-5x\right)+1$

$\frac{1}{3}\mathrm{log}\left(1-x\right)=\mathrm{log}\left(x+1\right)+\frac{1}{3}$

$x\approx -0.472$

## Extensions

Let $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ be any positive real number such that $\text{\hspace{0.17em}}b\ne 1.\text{\hspace{0.17em}}$ What must $\text{\hspace{0.17em}}{\mathrm{log}}_{b}1\text{\hspace{0.17em}}$ be equal to? Verify the result.

Explore and discuss the graphs of $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{\frac{1}{2}}\left(x\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=-{\mathrm{log}}_{2}\left(x\right).\text{\hspace{0.17em}}$ Make a conjecture based on the result.

The graphs of $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{\frac{1}{2}}\left(x\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)=-{\mathrm{log}}_{2}\left(x\right)\text{\hspace{0.17em}}$ appear to be the same; Conjecture: for any positive base $\text{\hspace{0.17em}}b\ne 1,$ $\text{\hspace{0.17em}}{\mathrm{log}}_{b}\left(x\right)=-{\mathrm{log}}_{\frac{1}{b}}\left(x\right).$

Prove the conjecture made in the previous exercise.

What is the domain of the function $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{ln}\left(\frac{x+2}{x-4}\right)?\text{\hspace{0.17em}}$ Discuss the result.

Recall that the argument of a logarithmic function must be positive, so we determine where $\text{\hspace{0.17em}}\frac{x+2}{x-4}>0\text{\hspace{0.17em}}$ . From the graph of the function $\text{\hspace{0.17em}}f\left(x\right)=\frac{x+2}{x-4},$ note that the graph lies above the x -axis on the interval $\text{\hspace{0.17em}}\left(-\infty ,-2\right)\text{\hspace{0.17em}}$ and again to the right of the vertical asymptote, that is $\text{\hspace{0.17em}}\left(4,\infty \right).\text{\hspace{0.17em}}$ Therefore, the domain is $\text{\hspace{0.17em}}\left(-\infty ,-2\right)\cup \left(4,\infty \right).$

Use properties of exponents to find the x -intercepts of the function $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{log}\left({x}^{2}+4x+4\right)\text{\hspace{0.17em}}$ algebraically. Show the steps for solving, and then verify the result by graphing the function.

The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris