6.3 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, $\ln (500) \approx 6.2146$

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Try It

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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Media

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

Introduction to Logarithms

Key equations

Definition of the logarithmic function	For $x > 0, b > 0, b \neq 1,$ $y = \log_{b} (x)$ 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 $b > 0, b \neq 1.$

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

If c is the cost function for a particular product, find the marginal cost functions and their values at x=10 a. c(x) = 800+ 0.04x + 0.0002x² b. c(x) = 250 + 100x + 0.001x²

Mamush Reply

how can I find set theory

Ephraim Reply

how can I find set theory

Jarvis

is there an error on the one about the dime's thickness? says 2.2x10⁶=0.00135 m

Patrick Reply

hi, interested in algebra

Makan Reply

how to reduce an equation?

Makan

by manipulation of both side

9(y+8)-27 is 9y+45. Why can't you reduce that to y+5? I know that's wrong but can't explain why

Patrick Reply

when you reduce an equation to its simplest terms, you can't change the value of the equation. reducing it to y + 5 is equivalent to dividing it by 9 which changes the value. you can multiply it by 1 or 9/9 which would give 9(y + 5). multiplying it by one does not change the value.

Philip

Given a polynomial expression, factor out the greatest common factor.

Hanu Reply

WHAT IS QUADRATIC EQUATION?

Charles Reply

WHAT IS SYSTEM OF LINEAR INEWUALITIES?

Charles

WHAT IS SYSTEM OF LINEAR INEWUALITIES?

Charles

complex perform

Angel

what is equation?

Charles Reply

what are equations?

Charles

Definition of economics according to karl Marx Thomas malthus Jeremy bentham David Ricardo J.K

Rakiya

Please help me is assignment

Rakiya

The 47th problem of Euclid

Kenneth

show that the set of all natural number form semi group under the composition of addition

Nikhil Reply

what is the meaning

Dominic

explain and give four Example hyperbolic function

Lukman Reply

_3_2_1

felecia

⅗ ⅔½

felecia

_½+⅔-¾

felecia

The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu

SABAL Reply

1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3

Pawel

2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31

Pawel

Ifeanyi

on number 2 question How did you got 2x +2

Ifeanyi

combine like terms. x + x + 2 is same as 2x + 2

Pawel

x*x=2

felecia

2+2x=

felecia

×/×+9+6/1

Debbie

Q2 x+(x+2)+(x+4)=60 3x+6=60 3x+6-6=60-6 3x=54 3x/3=54/3 x=18 :. The numbers are 18,20 and 22

Naagmenkoma

Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?

mariel Reply

Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113

Pawel

how do I set up the problem?

Harshika Reply

what is a solution set?

Harshika

find the subring of gaussian integers?

Rofiqul

hello, I am happy to help!

Shirley Reply

please can go further on polynomials quadratic

Abdullahi

hi mam

Mark

I need quadratic equation link to Alpa Beta

Abdullahi Reply

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Source: OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3

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