# 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

 $2^{5}=$ 4

 $5^{2}=$ Just like there is a way to say and write "4 to the power of 3" or "$4^{3}\!$ , there is a specific way to say and write logarithms.

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

However, it can also be written as:

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

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 $a,b,c,d,y>0$  with $b\neq 1,d\neq 1$ , we have

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

### Proof

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

$\log _{d}(b^{c})=\log _{d}(a)$ .

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

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

Isolating c on the left side gives

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

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

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

### Examples

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

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

Solve these logarithms

1

 $\log _{3}(81)=$ 2

 $\log _{6}(216)=$ 3

 $\log _{4}(64)=$ 4 Evaluate with a calculator (to 5dp)

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

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

 y=

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

 y=

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

 y=

### More properties

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

$5\times 6=30$

and

$30/6=5$

we also have

$7^{3}=343$

and

$\log _{7}343=3$

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

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

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

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

Logarithms are commonly abbreviated as logs.

### Ambiguity

The notation $\log x$  may refer to either $\ln x$  or $\log _{10}x$ , depending on the country and the context. For example, in English-speaking schools, $\log x$  usually means $\ln x$ , whereas it means $\log _{10}x$  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. $\log _{a}x+\log _{a}y=\log _{a}x*y$
2. $\log _{a}x-\log _{a}y=\log _{a}{\frac {x}{y}}$
3. $\log _{a}x^{b}=b\times \log _{a}x$

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

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

$\log _{a}x=b$  and $\log _{a}y=c$

$\ a^{b}=x$  and $\ a^{c}=y$

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

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

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

and replace b and c (as above)

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

## Change of Base Formula

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

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

$\ y^{b}=x$

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

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

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

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

Replace $\ b$  from first line

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

## Swap of Base and Exponent Formula

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

Proof:

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

Note that $a=c^{log_{c}a}$ . Then

$a^{log_{b}c}$  can be rewritten as

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

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

using the inverse rule noted above, this is equal to

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

and by the change of base formula

$c^{log_{b}a}$