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

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