# 4.3 Logarithmic functions  (Page 5/9)

 Page 5 / 9

## Evaluating a natural logarithm using a calculator

Evaluate $\text{\hspace{0.17em}}y=\mathrm{ln}\left(500\right)\text{\hspace{0.17em}}$ to four decimal places using a calculator.

• Press [LN] .
• Enter $\text{\hspace{0.17em}}500,$ followed by [ ) ] .
• Press [ENTER] .

Rounding to four decimal places, $\text{\hspace{0.17em}}\mathrm{ln}\left(500\right)\approx 6.2146$

Evaluate $\text{\hspace{0.17em}}\mathrm{ln}\left(-500\right).$

It is not possible to take the logarithm of a negative number in the set of real numbers.

Access this online resource for additional instruction and practice with logarithms.

## Key equations

 Definition of the logarithmic function For if and only if Definition of the common logarithm For if and only if Definition of the natural logarithm For if and only if

## Key concepts

• The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.
• Logarithmic equations can be written in an equivalent exponential form, using the definition of a logarithm. See [link] .
• Exponential equations can be written in their equivalent logarithmic form using the definition of a logarithm See [link] .
• Logarithmic functions with base $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ can be evaluated mentally using previous knowledge of powers of $\text{\hspace{0.17em}}b.\text{\hspace{0.17em}}$ See [link] and [link] .
• Common logarithms can be evaluated mentally using previous knowledge of powers of $\text{\hspace{0.17em}}10.\text{\hspace{0.17em}}$ See [link] .
• When common logarithms cannot be evaluated mentally, a calculator can be used. See [link] .
• Real-world exponential problems with base $\text{\hspace{0.17em}}10\text{\hspace{0.17em}}$ can be rewritten as a common logarithm and then evaluated using a calculator. See [link] .
• Natural logarithms can be evaluated using a calculator [link] .

## Verbal

What is a base $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ logarithm? Discuss the meaning by interpreting each part of the equivalent equations $\text{\hspace{0.17em}}{b}^{y}=x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{\mathrm{log}}_{b}x=y\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}b>0,b\ne 1.$

A logarithm is an exponent. Specifically, it is the exponent to which a base $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ is raised to produce a given value. In the expressions given, the base $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ has the same value. The exponent, $\text{\hspace{0.17em}}y,$ in the expression $\text{\hspace{0.17em}}{b}^{y}\text{\hspace{0.17em}}$ can also be written as the logarithm, $\text{\hspace{0.17em}}{\mathrm{log}}_{b}x,$ and the value of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is the result of raising $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ to the power of $\text{\hspace{0.17em}}y.$

How is the logarithmic function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}x\text{\hspace{0.17em}}$ related to the exponential function $\text{\hspace{0.17em}}g\left(x\right)={b}^{x}?\text{\hspace{0.17em}}$ What is the result of composing these two functions?

How can the logarithmic equation $\text{\hspace{0.17em}}{\mathrm{log}}_{b}x=y\text{\hspace{0.17em}}$ be solved for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ using the properties of exponents?

Since the equation of a logarithm is equivalent to an exponential equation, the logarithm can be converted to the exponential equation $\text{\hspace{0.17em}}{b}^{y}=x,$ and then properties of exponents can be applied to solve for $\text{\hspace{0.17em}}x.$

Discuss the meaning of the common logarithm. What is its relationship to a logarithm with base $\text{\hspace{0.17em}}b,$ and how does the notation differ?

Discuss the meaning of the natural logarithm. What is its relationship to a logarithm with base $\text{\hspace{0.17em}}b,$ and how does the notation differ?

The natural logarithm is a special case of the logarithm with base $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ in that the natural log always has base $\text{\hspace{0.17em}}e.\text{\hspace{0.17em}}$ Rather than notating the natural logarithm as $\text{\hspace{0.17em}}{\mathrm{log}}_{e}\left(x\right),$ the notation used is $\text{\hspace{0.17em}}\mathrm{ln}\left(x\right).$

## Algebraic

For the following exercises, rewrite each equation in exponential form.

${\text{log}}_{4}\left(q\right)=m$

${\text{log}}_{a}\left(b\right)=c$

${a}^{c}=b$

${\mathrm{log}}_{16}\left(y\right)=x$

${\mathrm{log}}_{x}\left(64\right)=y$

${x}^{y}=64$

${\mathrm{log}}_{y}\left(x\right)=-11$

#### Questions & Answers

I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
Â
Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
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
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
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
divide simplify each answer 3/2÷5/4
divide simplify each answer 25/3÷5/12
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
For Plan A to reach $27/month to surpass Plan B's$26.50 monthly payment, you'll need 3,000 texts which will cost an additional \$10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert