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Logarithms for Physics Help

By — McGraw-Hill Professional
Updated on Sep 17, 2011

Introduction

A logarithm (sometimes called a log ) is an exponent to which a constant is raised to obtain a given number. Suppose that the following relationship exists among three real numbers a , and x , and y :

a y = x

Then y is the base-a logarithm of x . The expression is written like this:

y = log a x

The two most often-used logarithm bases are 10 and e , where e an irrational number equal to approximately 2.71828.

Common Logs

Base-10 logarithms are also known as common logarithms or common logs . In equations, common logarithms are written as log without a subscript. For example:

log 10 = 1.000

Figure 5-1 is an approximate linear-coordinate graph of the function y = log x . Figure 5-2 is the same graph in semilog coordinates. The domain is limited to the positive real numbers. The range of the function encompasses the set of all real numbers.

Logarithms, Exponentials, and Trigonometry Logarithms Common Log In Terms Of Natural Log

Fig. 5-1 Approximate linear-coordinate graph of the common logarithm function.

Logarithms, Exponentials, and Trigonometry Logarithms Logarithm Of Product

Fig. 5-2 Approximate semilog-coordinate graph of the common logarithm function.

Natural Logs

Base- e logarithms are also called natural logs or napierian logs . In equations, the natural-log function is usually denoted ln or log e . For example:

ln 2.71828 = log e 2.71828 ≈ 1.00000

Figure 5-3 is an approximate linear-coordinate graph of the function y = ln x . Figure 5-4 is the same graph in semilog coordinates. The domain is limited to the positive real numbers, and the range spans the entire set of real numbers.

Logarithms, Exponentials, and Trigonometry Logarithms Logarithm Of Power

Fig. 5-3 Approximate linear-coordinate graph of the natural logarithm function.

 

Logarithms, Exponentials, and Trigonometry Logarithms Logarithm Of Root

Fig. 5-4 Approximate semilog-coordinate graph of the natural logarithm function.

Common Log In Terms Of Natural Log

Suppose that x is a positive real number. The common logarithm of x can be expressed in terms of the natural logarithms of x and 10:

log x = ln x /ln 10 ≈ 0.434 ln x

Natural Log In Terms Of Common Log

Suppose that x is a positive real number. The natural logarithm of x can be expressed in terms of the common logarithms of x and e:

ln x = log x /log e ≈ 2.303 log x

Logarithm Of Product

Suppose that x and y are both positive real numbers. The common or natural logarithm of the product is equal to the sum of the logarithms of the individual numbers:

log xy = log x + log y

ln xy = ln x + ln y

Logarithm Of Ratio

Let x and y be positive real numbers. The common or natural logarithm of their ratio, or quotient, is equal to the difference between the logarithms of the individual numbers:

log( x / y ) = log x − log y

ln( x / y ) = ln x − ln y

Logarithm Of Power

Suppose that x is a positive real number; let y be any real number. The common or natural logarithm of x raised to the power y can be reduced to a product as follows:

log x y = y log x

ln x y = y ln x

Logarithm Of Reciprocal

Suppose that x is a positive real number. The common or natural logarithm of the reciprocal (multiplicative inverse) of x is equal to the additive inverse of the logarithm of x :

log(1/ x ) = − log x

ln(1/ x ) = −ln x

Logarithm Of Root

Suppose that x is a positive real number and y is any real number except zero. The common or natural logarithm of the y th root of x (also denoted as x to the 1/ y power) can be found using the following equations:

log( x 1/ y ) = (log x )/ y

ln( x 1/ y ) = (ln x )/ y

Common Log Of Power Of 10

The common logarithm of 10 to any real-number power is always equal to that real number:

log(10 x ) = x

Natural Log Of Power Of E

The natural logarithm of e to any real-number power is always equal to that real number:

ln( e x ) = x

Practice problems for these concepts can be found at:  Logarithms, Exponentials, And Trigonometry for Physics Practice Test

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