1.1.4 Logarithms

FE Mechanical

1. Mathematics

1.1. Analytic geometry

Our FE Mechanical course is currently in development and is a work-in-progress.

Logarithms

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Logarithmic functions are among the most important mathematical tools in science and engineering. They serve as the inverse operations to exponential functions and provide a means to solve equations involving exponential expressions. Understanding logarithms is essential for applications ranging from measuring earthquake intensity to calculating compound interest and analyzing exponential growth and decay phenomena.

Definition of logarithms

The logarithm of a positive number $x$ to a positive base $b$ (where $b \neq = 1$ ) is defined as the exponent to which the base must be raised to produce that number:

$lo g_{b} (x) = c if and only if b^{c} = x$

This relationship establishes the fundamental connection between logarithmic and exponential functions. The logarithm answers the question: “To what power must we raise the base $b$ to obtain $x$ ?” A logarithm has three components:

Definitions

Base ( $b$ ): The number being raised to a power; must be positive and not equal to 1.
Argument ( $x$ ): The number whose logarithm we seek; must be positive.
Result ( $c$ ): The exponent or power to which the base is raised.

Special logarithmic bases

While logarithms can be defined for any valid base, two bases are of particular importance in mathematics and its applications.

Natural logarithm (base e)

The natural logarithm uses Euler’s number $e \approx 2.71828$ as its base:

$ln x = lo g_{e} x$

The natural logarithm appears in calculus, particularly in differentiation and integration of exponential functions. It is also fundamental in describing continuous growth and decay processes, including:

Calculus and differential equations
Continuous compound interest calculations
Population growth models
Radioactive decay
Normal distribution in statistics

Common logarithm (base 10)

When a logarithm expression is written without an explicit base, it typically represents base 10:

$lo g x = lo g_{10} x$

The common logarithm is particularly useful in scientific applications because our number system is decimal-based, including:

Scientific notation and order-of-magnitude calculations
pH scale in chemistry
Decibel scale for sound intensity
Richter scale for earthquake magnitude
Engineering and scientific measurements

Change of base

Often, it is necessary to convert logarithms from one base to another. The following formula provides a method for this conversion:

Change of base formula

$lo g_{b} x = \frac{lo g _{a} x}{lo g _{a} b}$

This formula allows us to express a logarithm in any base $b$ in terms of logarithms in any other base $a$ .

Logarithmic identities

Logarithmic identities are essential tools for simplifying expressions and solving equations. These identities follow directly from the properties of exponents.

Logarithms as inverse functions of exponential functions

$b^{l o g_{b} x} = x$
$lo g_{b} b^{x} = x$

Base and unity logarithms

$lo g_{b} b = 1$
$lo g_{b} 1 = 0$

In other words, the logarithm of the base itself is always 1, and the logarithm of 1 (in any base) is always 0.

Power rule for logarithms

$lo g_{b} (x^{c}) = c lo g_{b} x$

The power rule allows us to move exponents in and out of logarithms, which is particularly useful for solving exponential equations and simplifying complex expressions.

Product and quotient rules for logarithms

$lo g_{b} (x y) = lo g_{b} x + lo g_{b} y$

This rule states that the logarithm of a product equals the sum of the logarithms of the factors.

$lo g_{b} (\frac{x}{y}) = lo g_{b} x - lo g_{b} y$

This rule states that the logarithm of a quotient equals the difference of the logarithms.

Linearizing exponential data

In scientific applications, logarithms are used to linearize exponential relationships, making them easier to analyze and fit to data. If $y = a b^{x}$ , then:

$lo g y = lo g a + x lo g b$

This transforms the exponential relationship into a linear one in terms of the new dependent variable $lo g y$ and independent variable $x$ .

Sidenote

Common mistakes and misconceptions

The logarithm of the sum of two numbers is not the sum of their logarithms:

Incorrect: $lo g (x + y) = lo g x + lo g y$ .
Correct: There is no simple rule for $lo g (x + y)$ .

Definition of logarithms:

$lo g_{b} (x) = c if and only if b^{c} = x$

Natural logarithm (Base $e$ ):

$ln x = lo g_{e} x$

Common logarithm (Base 10):

$lo g x = lo g_{10} x$

Change of base Formula:

$lo g_{b} x = \frac{lo g _{a} x}{lo g _{a} b}$

Base & unity logarithms:

$lo g_{b} b = 1$

$lo g_{b} 1 = 0$

Power rule for logarithms:

$lo g_{b} (x^{c}) = c lo g_{b} x$

Product & quotient rules for logarithms:

$lo g_{b} (x y) = lo g_{b} x + lo g_{b} y$

$lo g_{b} (\frac{x}{y}) = lo g_{b} x - lo g_{b} y$

Next | 1.2.1 Differential calculus

Logarithms

Definition of logarithms

The logarithm of a positive number $x$ to a positive base $b$ (where $b \neq = 1$ ) is defined as the exponent to which the base must be raised to produce that number:

$lo g_{b} (x) = c if and only if b^{c} = x$

Definitions

Base ( $b$ ): The number being raised to a power; must be positive and not equal to 1.
Argument ( $x$ ): The number whose logarithm we seek; must be positive.
Result ( $c$ ): The exponent or power to which the base is raised.

Special logarithmic bases

While logarithms can be defined for any valid base, two bases are of particular importance in mathematics and its applications.

Natural logarithm (base e)

The natural logarithm uses Euler’s number $e \approx 2.71828$ as its base:

$ln x = lo g_{e} x$

Calculus and differential equations
Continuous compound interest calculations
Population growth models
Radioactive decay
Normal distribution in statistics

Common logarithm (base 10)

When a logarithm expression is written without an explicit base, it typically represents base 10:

$lo g x = lo g_{10} x$

The common logarithm is particularly useful in scientific applications because our number system is decimal-based, including:

Scientific notation and order-of-magnitude calculations
pH scale in chemistry
Decibel scale for sound intensity
Richter scale for earthquake magnitude
Engineering and scientific measurements

Change of base

Often, it is necessary to convert logarithms from one base to another. The following formula provides a method for this conversion:

Change of base formula

$lo g_{b} x = \frac{lo g _{a} x}{lo g _{a} b}$

This formula allows us to express a logarithm in any base $b$ in terms of logarithms in any other base $a$ .

Logarithmic identities

Logarithmic identities are essential tools for simplifying expressions and solving equations. These identities follow directly from the properties of exponents.

Logarithms as inverse functions of exponential functions

$b^{l o g_{b} x} = x$
$lo g_{b} b^{x} = x$

Base and unity logarithms

$lo g_{b} b = 1$
$lo g_{b} 1 = 0$

In other words, the logarithm of the base itself is always 1, and the logarithm of 1 (in any base) is always 0.

Power rule for logarithms

$lo g_{b} (x^{c}) = c lo g_{b} x$

The power rule allows us to move exponents in and out of logarithms, which is particularly useful for solving exponential equations and simplifying complex expressions.

Product and quotient rules for logarithms

$lo g_{b} (x y) = lo g_{b} x + lo g_{b} y$

This rule states that the logarithm of a product equals the sum of the logarithms of the factors.

$lo g_{b} (\frac{x}{y}) = lo g_{b} x - lo g_{b} y$

This rule states that the logarithm of a quotient equals the difference of the logarithms.

Linearizing exponential data

In scientific applications, logarithms are used to linearize exponential relationships, making them easier to analyze and fit to data. If $y = a b^{x}$ , then:

$lo g y = lo g a + x lo g b$

This transforms the exponential relationship into a linear one in terms of the new dependent variable $lo g y$ and independent variable $x$ .

Sidenote

Common mistakes and misconceptions

The logarithm of the sum of two numbers is not the sum of their logarithms:

Incorrect: $lo g (x + y) = lo g x + lo g y$ .
Correct: There is no simple rule for $lo g (x + y)$ .

FE Mechanical

1. Mathematics

1.1. Analytic geometry

Our FE Mechanical course is currently in development and is a work-in-progress.

Logarithms

4 min read

Font

Discuss

Feedback

Definition of logarithms

The logarithm of a positive number $x$ to a positive base $b$ (where $b \neq = 1$ ) is defined as the exponent to which the base must be raised to produce that number:

$lo g_{b} (x) = c if and only if b^{c} = x$

Definitions

Base ( $b$ ): The number being raised to a power; must be positive and not equal to 1.
Argument ( $x$ ): The number whose logarithm we seek; must be positive.
Result ( $c$ ): The exponent or power to which the base is raised.

Special logarithmic bases

While logarithms can be defined for any valid base, two bases are of particular importance in mathematics and its applications.

Natural logarithm (base e)

The natural logarithm uses Euler’s number $e \approx 2.71828$ as its base:

$ln x = lo g_{e} x$

Calculus and differential equations
Continuous compound interest calculations
Population growth models
Radioactive decay
Normal distribution in statistics

Common logarithm (base 10)

When a logarithm expression is written without an explicit base, it typically represents base 10:

$lo g x = lo g_{10} x$

The common logarithm is particularly useful in scientific applications because our number system is decimal-based, including:

Scientific notation and order-of-magnitude calculations
pH scale in chemistry
Decibel scale for sound intensity
Richter scale for earthquake magnitude
Engineering and scientific measurements

Change of base

Often, it is necessary to convert logarithms from one base to another. The following formula provides a method for this conversion:

Change of base formula

$lo g_{b} x = \frac{lo g _{a} x}{lo g _{a} b}$

This formula allows us to express a logarithm in any base $b$ in terms of logarithms in any other base $a$ .

Logarithmic identities

Logarithmic identities are essential tools for simplifying expressions and solving equations. These identities follow directly from the properties of exponents.

Logarithms as inverse functions of exponential functions

$b^{l o g_{b} x} = x$
$lo g_{b} b^{x} = x$

Base and unity logarithms

$lo g_{b} b = 1$
$lo g_{b} 1 = 0$

In other words, the logarithm of the base itself is always 1, and the logarithm of 1 (in any base) is always 0.

Power rule for logarithms

$lo g_{b} (x^{c}) = c lo g_{b} x$

The power rule allows us to move exponents in and out of logarithms, which is particularly useful for solving exponential equations and simplifying complex expressions.

Product and quotient rules for logarithms

$lo g_{b} (x y) = lo g_{b} x + lo g_{b} y$

This rule states that the logarithm of a product equals the sum of the logarithms of the factors.

$lo g_{b} (\frac{x}{y}) = lo g_{b} x - lo g_{b} y$

This rule states that the logarithm of a quotient equals the difference of the logarithms.

Linearizing exponential data

In scientific applications, logarithms are used to linearize exponential relationships, making them easier to analyze and fit to data. If $y = a b^{x}$ , then:

$lo g y = lo g a + x lo g b$

This transforms the exponential relationship into a linear one in terms of the new dependent variable $lo g y$ and independent variable $x$ .

Sidenote

Common mistakes and misconceptions

The logarithm of the sum of two numbers is not the sum of their logarithms:

Incorrect: $lo g (x + y) = lo g x + lo g y$ .
Correct: There is no simple rule for $lo g (x + y)$ .

Definition of logarithms:

$lo g_{b} (x) = c if and only if b^{c} = x$

Natural logarithm (Base $e$ ):

$ln x = lo g_{e} x$

Common logarithm (Base 10):

$lo g x = lo g_{10} x$

Change of base Formula:

$lo g_{b} x = \frac{lo g _{a} x}{lo g _{a} b}$

Base & unity logarithms:

$lo g_{b} b = 1$

$lo g_{b} 1 = 0$

Power rule for logarithms:

$lo g_{b} (x^{c}) = c lo g_{b} x$

Product & quotient rules for logarithms:

$lo g_{b} (x y) = lo g_{b} x + lo g_{b} y$

$lo g_{b} (\frac{x}{y}) = lo g_{b} x - lo g_{b} y$

Next | 1.2.1 Differential calculus

Logarithms

Definition of logarithms

The logarithm of a positive number $x$ to a positive base $b$ (where $b \neq = 1$ ) is defined as the exponent to which the base must be raised to produce that number:

$lo g_{b} (x) = c if and only if b^{c} = x$

Definitions

Base ( $b$ ): The number being raised to a power; must be positive and not equal to 1.
Argument ( $x$ ): The number whose logarithm we seek; must be positive.
Result ( $c$ ): The exponent or power to which the base is raised.

Special logarithmic bases

While logarithms can be defined for any valid base, two bases are of particular importance in mathematics and its applications.

Natural logarithm (base e)

The natural logarithm uses Euler’s number $e \approx 2.71828$ as its base:

$ln x = lo g_{e} x$

Calculus and differential equations
Continuous compound interest calculations
Population growth models
Radioactive decay
Normal distribution in statistics

Common logarithm (base 10)

When a logarithm expression is written without an explicit base, it typically represents base 10:

$lo g x = lo g_{10} x$

The common logarithm is particularly useful in scientific applications because our number system is decimal-based, including:

Scientific notation and order-of-magnitude calculations
pH scale in chemistry
Decibel scale for sound intensity
Richter scale for earthquake magnitude
Engineering and scientific measurements

Change of base

Often, it is necessary to convert logarithms from one base to another. The following formula provides a method for this conversion:

Change of base formula

$lo g_{b} x = \frac{lo g _{a} x}{lo g _{a} b}$

This formula allows us to express a logarithm in any base $b$ in terms of logarithms in any other base $a$ .

Logarithmic identities

Logarithmic identities are essential tools for simplifying expressions and solving equations. These identities follow directly from the properties of exponents.

Logarithms as inverse functions of exponential functions

$b^{l o g_{b} x} = x$
$lo g_{b} b^{x} = x$

Base and unity logarithms

$lo g_{b} b = 1$
$lo g_{b} 1 = 0$

In other words, the logarithm of the base itself is always 1, and the logarithm of 1 (in any base) is always 0.

Power rule for logarithms

$lo g_{b} (x^{c}) = c lo g_{b} x$

The power rule allows us to move exponents in and out of logarithms, which is particularly useful for solving exponential equations and simplifying complex expressions.

Product and quotient rules for logarithms

$lo g_{b} (x y) = lo g_{b} x + lo g_{b} y$

This rule states that the logarithm of a product equals the sum of the logarithms of the factors.

$lo g_{b} (\frac{x}{y}) = lo g_{b} x - lo g_{b} y$

This rule states that the logarithm of a quotient equals the difference of the logarithms.

Linearizing exponential data

In scientific applications, logarithms are used to linearize exponential relationships, making them easier to analyze and fit to data. If $y = a b^{x}$ , then:

$lo g y = lo g a + x lo g b$

This transforms the exponential relationship into a linear one in terms of the new dependent variable $lo g y$ and independent variable $x$ .

Sidenote

Common mistakes and misconceptions

The logarithm of the sum of two numbers is not the sum of their logarithms:

Incorrect: $lo g (x + y) = lo g x + lo g y$ .
Correct: There is no simple rule for $lo g (x + y)$ .