# Algebra/Chapter 12/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. It basically answers the question: what power do I have to raise this base to, to get this result.

## Logarithms

In order to understand logs, we need to review exponential equations. Answer the following problems:
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
* $\log _{b}(y^{a})=a\log _{b}(y)$
* $\log _{b}(b^{a})=a$
* $\log _{b}(ac)=\log _{b}(a)+\log _{b}(c)$
* $\log _{b}(a/c)=\log _{b}(a)-\log _{b}(c)$
* $\log _{b}(a)={\frac {\log _{d}(a)}{\log _{d}(b)}}\quad$ (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,
    $\log _{3}(12)={\frac {\log _{10}(12)}{\log _{10}(3)}}=2.262$

### 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

* $\log _{a}x+\log _{a}y=\log _{a}x*y$
* $\log _{a}x-\log _{a}y=\log _{a}{\frac {x}{y}}$
* $\log _{a}x^{b}=b\times \log _{a}x$
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
$c^{log_{b}a}$