**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.

**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.

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

**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*

*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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