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4.3 Logarithmic functions  (Page 5/9)

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Evaluating a natural logarithm using a calculator

Evaluate   y = ln ( 500 )   to four decimal places using a calculator.

  • Press [LN] .
  • Enter   500 , followed by [ ) ] .
  • Press [ENTER] .

Rounding to four decimal places,   ln ( 500 ) ≈ 6.2146

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Evaluate   ln ( −500 ) .

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

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

Key equations

Definition of the logarithmic function For     x > 0 , b > 0 , b ≠ 1 ,
y = log b ( x )   if and only if   b y = x .
Definition of the common logarithm For   x > 0 , y = log ( x )   if and only if   10 y = x .
Definition of the natural logarithm For   x > 0 , y = ln ( x )   if and only if   e y = x .

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   b   can be evaluated mentally using previous knowledge of powers of   b .   See [link] and [link] .
  • Common logarithms can be evaluated mentally using previous knowledge of powers of   10.   See [link] .
  • When common logarithms cannot be evaluated mentally, a calculator can be used. See [link] .
  • Real-world exponential problems with base   10   can be rewritten as a common logarithm and then evaluated using a calculator. See [link] .
  • Natural logarithms can be evaluated using a calculator [link] .

Section exercises

Verbal

What is a base   b   logarithm? Discuss the meaning by interpreting each part of the equivalent equations   b y = x   and   log b x = y   for   b > 0 , b ≠ 1.

A logarithm is an exponent. Specifically, it is the exponent to which a base   b   is raised to produce a given value. In the expressions given, the base   b   has the same value. The exponent,   y , in the expression   b y   can also be written as the logarithm,   log b x , and the value of   x   is the result of raising   b   to the power of   y .

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How is the logarithmic function   f ( x ) = log b x   related to the exponential function   g ( x ) = b x ?   What is the result of composing these two functions?

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How can the logarithmic equation   log b x = y   be solved for   x   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   b y = x , and then properties of exponents can be applied to solve for   x .

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Discuss the meaning of the common logarithm. What is its relationship to a logarithm with base   b , and how does the notation differ?

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Discuss the meaning of the natural logarithm. What is its relationship to a logarithm with base   b , and how does the notation differ?

The natural logarithm is a special case of the logarithm with base   b   in that the natural log always has base   e .   Rather than notating the natural logarithm as   log e ( x ) , the notation used is   ln ( x ) .

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Algebraic

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

log 4 ( q ) = m

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log a ( b ) = c

a c = b

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log 16 ( y ) = x

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log x ( 64 ) = y

x y = 64

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log y ( x ) = −11

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Questions & Answers

for the "hiking" mix, there are 1,000 pieces in the mix, containing 390.8 g of fat, and 165 g of protein. if there is the same amount of almonds as cashews, how many of each item is in the trail mix?
ADNAN Reply
linear speed of an object
Melissa Reply
an object is traveling around a circle with a radius of 13 meters .if in 20 seconds a central angle of 1/7 Radian is swept out what are the linear and angular speed of the object
Melissa
test
Matrix
how to find domain
Mohamed Reply
like this: (2)/(2-x) the aim is to see what will not be compatible with this rational expression. If x= 0 then the fraction is undefined since we cannot divide by zero. Therefore, the domain consist of all real numbers except 2.
Dan
define the term of domain
Moha
if a>0 then the graph is concave
Angel Reply
if a<0 then the graph is concave blank
Angel
what's a domain
Kamogelo Reply
The set of all values you can use as input into a function su h that the output each time will be defined, meaningful and real.
Spiro
how fast can i understand functions without much difficulty
Joe Reply
what is inequalities
Nathaniel
functions can be understood without a lot of difficulty. Observe the following: f(2) 2x - x 2(2)-2= 2 now observe this: (2,f(2)) ( 2, -2) 2(-x)+2 = -2 -4+2=-2
Dan
what is set?
Kelvin Reply
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
158.5 This number can be developed by using algebra and logarithms. Begin by moving log(2) to the right hand side of the equation like this: t/100 log(2)= log(3) step 1: divide each side by log(2) t/100=1.58496250072 step 2: multiply each side by 100 to isolate t. t=158.49
Dan
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
how to find x: 12x = 144 notice how 12 is being multiplied by x. Therefore division is needed to isolate x and whatever we do to one side of the equation we must do to the other. That develops this: x= 144/12 divide 144 by 12 to get x. addition: 12+x= 14 subtract 12 by each side. x =2
Dan
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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