# Algebra/Logarithms

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

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

 ${\displaystyle 2^{5}=}$

4

 ${\displaystyle 5^{2}=}$

Just like there is a way to say and write "4 to the power of 3" or "${\displaystyle 4^{3}\!}$ , there is a specific way to say and write logarithms.

For example, "4 to the power of 3 equals 64" can be written as: ${\displaystyle 4^{3}=64\!}$

However, it can also be written as:

${\displaystyle \log _{4}(64)=3\ }$

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: ${\displaystyle 2^{5}=32\!}$

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

The following properties derive from the definition of logarithm.

### Basic properties

For all real numbers ${\displaystyle a,b,c,d,y>0}$  with ${\displaystyle b\neq 1,d\neq 1}$ , we have

1. ${\displaystyle \log _{b}(y^{a})=a\log _{b}(y)}$
2. ${\displaystyle \log _{b}(b^{a})=a}$
3. ${\displaystyle \log _{b}(ac)=\log _{b}(a)+\log _{b}(c)}$
4. ${\displaystyle \log _{b}(a/c)=\log _{b}(a)-\log _{b}(c)}$
5. ${\displaystyle \log _{b}(a)={\frac {\log _{d}(a)}{\log _{d}(b)}}\quad }$  (change of base rule).

### Proof

Let us take the log to base d of both sides of the equation ${\displaystyle b^{c}=a}$ :

${\displaystyle \log _{d}(b^{c})=\log _{d}(a)}$ .

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

${\displaystyle c\log _{d}(b)=\log _{d}(a)}$

Isolating c on the left side gives

${\displaystyle c={\frac {\log _{d}(a)}{\log _{d}(b)}}}$

Finally, since ${\displaystyle c=\log _{b}(a)}$

${\displaystyle \log _{b}(a)={\frac {\log _{d}(a)}{\log _{d}(b)}}}$

### Examples

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

${\displaystyle \log _{3}(12)={\frac {\log _{10}(12)}{\log _{10}(3)}}=2.262}$

Solve these logarithms

1

 ${\displaystyle \log _{3}(81)=}$

2

 ${\displaystyle \log _{6}(216)=}$

3

 ${\displaystyle \log _{4}(64)=}$

4 Evaluate with a calculator (to 5dp)

 ${\displaystyle \log _{4}(6)=}$
Find the y value of these logarithms

5 ${\displaystyle \log _{3}(y)=3}$

 y=

6 ${\displaystyle \log _{5}(y)=4}$

 y=

7 ${\displaystyle \log _{9}(y)=4}$

 y=

### More properties

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

${\displaystyle 5\times 6=30}$

and

${\displaystyle 30/6=5}$

we also have

${\displaystyle 7^{3}=343}$

and

${\displaystyle \log _{7}343=3}$

More generally, if ${\displaystyle a^{b}=x}$ , then ${\displaystyle \log _{a}x=b}$ . Also, if ${\displaystyle f(x)=a^{x}}$ , then ${\displaystyle f^{-1}(x)=\log _{a}x}$ , so if the two equations are graphed, each one is the reflection of the other over the line ${\displaystyle y=x}$ . (In both equations, a is called the base.)

As a result, ${\displaystyle a^{\log _{a}b}=b}$  and ${\displaystyle \log _{a}a^{b}=b}$ .

Common bases for logarithms are the base of 10 (${\displaystyle \log _{10}x}$  is known as the common logarithm) and the base e (${\displaystyle \ln x}$  is known as the natural logarithm), where e = 2.71828182846...

Natural logs are usually written as ${\displaystyle \ln x}$  or ${\displaystyle \ln(x)}$  (ln is short for natural logarithm in Latin), and sometimes as ${\displaystyle \log _{e}x}$  or ${\displaystyle \log _{e}(x)}$ . Parenthesized forms are recommended when x is a mathematical expression (e.g., ${\displaystyle \ln(6x+1)}$ ).

Logarithms are commonly abbreviated as logs.

### Ambiguity

The notation ${\displaystyle \log x}$  may refer to either ${\displaystyle \ln x}$  or ${\displaystyle \log _{10}x}$ , depending on the country and the context. For example, in English-speaking schools, ${\displaystyle \log x}$  usually means ${\displaystyle \ln x}$ , whereas it means ${\displaystyle \log _{10}x}$  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

1. ${\displaystyle \log _{a}x+\log _{a}y=\log _{a}x*y}$
2. ${\displaystyle \log _{a}x-\log _{a}y=\log _{a}{\frac {x}{y}}}$
3. ${\displaystyle \log _{a}x^{b}=b\times \log _{a}x}$

Proof:
${\displaystyle \log _{a}x+\log _{a}y=\log _{a}x*y}$

${\displaystyle \log _{a}x+\log _{a}y}$

${\displaystyle \log _{a}x=b}$  and ${\displaystyle \log _{a}y=c}$

${\displaystyle \ a^{b}=x}$  and ${\displaystyle \ a^{c}=y}$

${\displaystyle \ xy=a^{b}a^{c}}$

${\displaystyle \ xy=a^{(b+c)}}$

${\displaystyle \log _{a}xy=b+c}$

and replace b and c (as above)

${\displaystyle \log _{a}xy=\log _{a}x+\log _{a}y}$

## Change of Base Formula

${\displaystyle \log _{y}x={\frac {\log _{a}x}{\log _{a}y}}}$  where a is any positive number, distinct from 1. Generally, a is either 10 (for common logs) or e (for natural logs).

Proof:
${\displaystyle \log _{y}x=b}$

${\displaystyle \ y^{b}=x}$

Put both sides to ${\displaystyle \log _{a}}$

${\displaystyle \log _{a}y^{b}=\log _{a}x}$

${\displaystyle \ b\log _{a}y=\log _{a}x}$

${\displaystyle \ b={\frac {\log _{a}x}{\log _{a}y}}}$

Replace ${\displaystyle \ b}$  from first line

${\displaystyle \log _{y}x={\frac {\log _{a}x}{\log _{a}y}}}$

## Swap of Base and Exponent Formula

${\displaystyle a^{\log _{b}c}=c^{\log _{b}a}}$  where a or c must not be equal to 1.

Proof:

${\displaystyle log_{a}b={\frac {1}{log_{b}a}}}$  by the change of base formula above.

Note that ${\displaystyle a=c^{log_{c}a}}$ . Then

${\displaystyle a^{log_{b}c}}$  can be rewritten as

${\displaystyle ({c^{log_{c}a}})^{log_{b}c}}$  or by the exponential rule as

${\displaystyle c^{{log_{c}a}*{log_{b}c}}}$

using the inverse rule noted above, this is equal to

${\displaystyle c^{{log_{c}a}*{\frac {1}{log_{c}b}}}}$

and by the change of base formula

${\displaystyle c^{log_{b}a}}$