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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 x ( 64 ) = y

x y = 64

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Questions & Answers

By the definition, is such that 0!=1.why?
Unikpel Reply
(1+cosA+IsinA)(1+cosB+isinB)/(cos@+isin@)(cos$+isin$)
Ajay Reply
hatdog
Mark
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given that x= 3/5 find sin 3x
Adamu Reply
4
DB
remove any signs and collect terms of -2(8a-3b-c)
Joeval Reply
-16a+6b+2c
Will
is that a real answer
Joeval
(x2-2x+8)-4(x2-3x+5)
Ayush Reply
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Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
X2-2X+8-4X2+12X-20=0 (X2-4X2)+(-2X+12X)+(-20+8)= 0 -3X2+10X-12=0 3X2-10X+12=0 Use quadratic formula To find the answer answer (5±Root11i)/3
master
Soo sorry (5±Root11* i)/3
master
x2-2x+8-4x2+12x-20 x2-4x2-2x+12x+8-20 -3x2+10x-12 now you can find the answer using quadratic
Mukhtar
2x²-6x+1=0
Ife
explain and give four example of hyperbolic function
Lukman Reply
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
Racelle Reply
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Anurag Reply
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
y2=4ax= y=4ax/2. y=2ax
akash
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
Jhovie Reply
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
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Charmaine Reply
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Daniel
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Miranda
Practice Key Terms 3

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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