# 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) ${\displaystyle 5^{2}\!}$

1: "4 to the power of 3" means "4 multiplied 3 times." Thus, 4 to the power of 3 equals 4 times 4 times 4. 4 times 4=16. 16 times 4=64. So 4 to the power of 3 equals 64.

2: 3 times 3 = 9. Times 3 equals 27. Times 3 equals 81. So 3 to the power of 4 equals 81.

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

4: ${\displaystyle 5^{2}=5*5=25\!}$

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 properities derive from the definition of logarithm.

If ${\displaystyle b>0}$ with ${\displaystyle b<>1}$, then for every real y,a,c it is:

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

There is also the "change of base rule":

${\displaystyle \log _{b}(a)={\frac {\log _{d}(a)}{\log _{d}(b)}}}$ for any ${\displaystyle d>0,d\neq 1}$

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

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

This rule allows you 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
• ${\displaystyle \log _{3}(81)=\!}$
• ${\displaystyle \log _{6}(216)=\!}$
• ${\displaystyle \log _{4}(64)=\!}$
• Evaluate with a calculator
• ${\displaystyle \log _{4}(6)=\!}$
• Find the y value of these logarithms
• ${\displaystyle \log _{3}(y)=3\!}$
• ${\displaystyle \log _{5}(y)=4\!}$
• ${\displaystyle \log _{9}(y)=4\!}$

• Solve these logarithms
• 4
• 3
• 3
• Evaluate with a calculator
• 1.29248
• Find the y value of these logarithms
• 27
• 625
• 6561

Logarithms are the reverse of exponential functions, just as division is the reverse of multiplication, for example:

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

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

Or, in a more general form, 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 is the reflection of the other over the line ${\displaystyle y=x}$. (in both equations, a is considered to be the base).

Because of this, ${\displaystyle a^{\log _{a}b}=b}$ and ${\displaystyle \log _{a}a^{b}=b}$.

Common bases used are bases of 10 which is a called a common logarithm or e which is called a natural logarithm" (e~=2.71828182846).

Common logs are written either as ${\displaystyle \log _{10}x}$ or simply as ${\displaystyle \log x}$.

Natural logs are written either as ${\displaystyle \log _{e}x}$ or simply as ${\displaystyle \ln x}$ (the ln stands for natural logarithm).

Logarithms are commonly abbreviated as logs.

## 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

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