Logarithms

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Basic Description

Logarithms are considered the inverse or opposite operations to exponents, just as subtraction is the inverse to addition or square rooting is the inverse to squaring.


For example, suppose we have the exponential expression 2^3 , which we know will equal 8. Now, suppose we want to do the inverse operation and go from the value 8 to the exponent 3 with a base of 2. We could do the inverse operation by using logarithms and write  log_{2}(8) = 3 , which is read "logarithm of 8 base 2 is equal to 3".


In order words, if we have an exponential equation: base^x = n\,
we can write an equivalent logarithmic equation: log_{base}(n) = x\,


What about exponential equations such as 10^x = 932? It might seem harder to solve for x in this case because there is no whole number exponent that will give us the value of 932 with a base of 10. However, if we simply rewrite the equation as an logarithmic equation x = log_{10}(932) , we can find quite easily with a calculator that x is about 2.969.


To look at some more examples of switching between exponential and logarithmic equations:

Exponential Equation Logarithmic Equation
7^3 = 343  \, Answer
Answer 6 = log_{3}(729)\,
Answer 4 = log_{10}(10000)\,
e^7 \approx 1096.6  \, Answer
Answer 3.5 \approx log_{6}(529.1)\,
22^x = 57643  \, Answer

A More Mathematical Description

Definition of a Logarithm

 y = log_a(x)\,   if and only if   x = a^y\,,      where b > 1 and x > 0
 In words: The logarithm of a value at a given base is the power (exponent) that the base must be raised to produce the value.

Bases

As seen from the definition above, the base of a logarithm affects how a logarithm is evaluated. Bases can be any positive number except for 1, and the logarithms of a value can be found at different bases using a change of base formula, as shown in the section below the Common Bases chart.


Common Bases

There are three main bases that are most frequently used:

Base Exponential Representation Logarithmic Representation Notes Example
Base 10 y = 10^x\, x = log_{10}(y)\, can be written simply as log(x)\,

also called Common Logarithms

100 = 10^x\,

x = log(100)\,

where x = 2
Base 2 y = 2^x\, x = log_2(y)\, basis for the Binary System 16 = 2^x\,

x = log_{2}(16)\,

where x = 4
Base e y = e^x\, x = log_{e}(y)\, can be written simply as ln(y)\,

also called Natural Logarithms

where ln(e) = 1\,
25 = e^x\,

x = ln(25)\,

where x \approx 3.22


Changing Bases

To go from a logarithm of base k to a logarithm of base a, we use the formula:

\frac{log_k(x)}{log_k(a)} = log_a(x) \,

x = a^n \,
log_k(x) = log_k(a^n) \, Taking the log_k of both sides
log_k(x) = nlog_k(a) \, Using the exponential property of logarithms
\frac{log_k(x)}{log_k(a)} = n \,Dividing by log_k(a)
\frac{log_k(x)}{log_k(a)} = log_a(x) \,From a logarithm's definition x = a^n and thus  n = log_a(x)

For example, let us take the logarithm of 14 to the base 10, which we are given, and find the logarithm of 14 to the base 2, which we do not know.

We would designate k = 10, a = 2, and x = 14.


log_a(x)=\frac{log_k(x)}{log_k(a)}


log_{2}(14)=\frac{log_{(10)}(14)}{log_{(10)}(12)}


log_2(14) \approx \frac{1.146}{0.301}


log_2(14) \approx 3.807


Graphing a Logarithmic Function

The image on the left is a graph of a basic logarithmic function: y = log(x)\,. A logarithmic function, such as the one used to create the featured image, takes the basic form y = log_b(x)\,, where b is fixed while y and x are variables. In addition, there is an vertical asymptote at x = 0, so that logarithmic functions are undefined when x is less than or equal to 0. However, non-real logarithms for negative x values can be found using complex logarithms with complex numbers.


The second image compares the graphical representations of the most common logarithmic functions: y = log(x)\,, y = log_2(x)\,, and y = log_e(x)\,.


Basic Properties of Logarithms

Logarithms possess various properties and identities including the following:

Identities
Multiplication log(x * y) = log(x) + log(y) \,
Division log\left(\frac{x}{y}\right) = log(x) - log(y) \,
Exponentiation log(x^n) = nlog(x)\,
Integration \int log(x) = xlog(x) - x + constant\,
Differentiation \frac{d}{dx}(log(x)) = \frac{1}{x} \,
Other properties
log(1) = 0\,
log(0) = undefined\,
log_a(a^x) = x\,
 a^{log_a(x)} = x \,
log_a(b) = \frac{1}{log_b(a)}\,



Additional Resources

Reference used - Wikipedia, Logarithms Page
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