# 6.3 Logarithmic functions  (Page 5/9)

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## 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

what is math number
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
hi vedant can u help me with some assignments
Solomon
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
sure. what is your question?
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar