Algebra/Logarithms

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Logarithms (commonly called "logs") are a specific instance of a function being used for everyday use. Logarithms are used commonly to measure earthquakes, distances of stars, economics, and throughout the scientific world.

Logarithms[edit]

In order to understand logs, we need to review exponential equations. Answer the following problems:

1 What is 4 to the power of 3?

2 What is 3 to the power of 4?

3

4


Just like there is a way to say and write "4 to the power of 3" or ", there is a specific way to say and write logarithms.

For example, "4 to the power of 3 equals 64" can be written as:

However, it can also be written as:

Once, you remember that the base of the exponent is the number being raised to a power and that the base of the logarithm is the subscript after the log, the rest falls into place. I like to draw an arrow (either mentally or physically) from the base, to the exponent, to the product when changing from logarithmic form to exponential form. So visually or mentally I would go from 2 to 5 to 32 in the logarithmic example which (once I add the conventions) gives us:

So, when you are given a logarithm to solve, just remember how to convert it to an exponential equation. Here are some practice problems, the answers are at the bottom.

Properties of Logarithms[edit]

The following properties derive from the definition of logarithm.

Basic properties[edit]

For all real numbers with , we have

  1. (change of base rule).

Proof[edit]

Let us take the log to base d of both sides of the equation :

.

Next, notice that the left side of this equation is the same as that in property number 1 above. Let us apply this property:

Isolating c on the left side gives

Finally, since

Examples[edit]

This rule allows us to evaluate logs to a base other than e or 10 on a calculator. For example,

Solve these logarithms

1

2

3

4 Evaluate with a calculator (to 5dp)

Find the y value of these logarithms

5

y=

6

y=

7

y=


More properties[edit]

Logarithms are the reverse of exponential functions, just as division is the reverse of multiplication. For example, just as we have

and

we also have

and

More generally, if , then . Also, if , then , so if the two equations are graphed, each one is the reflection of the other over the line . (In both equations, a is called the base.)

As a result, and .

Common bases for logarithms are the base of 10 ( is known as the common logarithm) and the base e ( is known as the natural logarithm), where e = 2.71828182846...

Natural logs are usually written as or (ln is short for natural logarithm in Latin), and sometimes as or . Parenthesized forms are recommended when x is a mathematical expression (e.g., ).

Logarithms are commonly abbreviated as logs.

Ambiguity[edit]

The notation may refer to either or , depending on the country and the context. For example, in English-speaking schools, usually means , whereas it means in Italian- and French-speaking schools or to English-speaking number theorists. Consequently, this notation should only be used when the context is clear.

Properties of Logarithms[edit]

Proof:

and

and

and replace b and c (as above)

Change of Base Formula[edit]

where a is any positive number, distinct from 1. Generally, a is either 10 (for common logs) or e (for natural logs).

Proof:

Put both sides to

Replace from first line

Swap of Base and Exponent Formula[edit]

where a or c must not be equal to 1.

Proof:

by the change of base formula above.

Note that . Then

can be rewritten as

or by the exponential rule as

using the inverse rule noted above, this is equal to

and by the change of base formula