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 properities derive from the definition of logarithm.

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

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)\$

There is also the "change of base rule":

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

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

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

This rule allows you 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=

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

$5\times 6=30$ and $30/6=5$

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

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

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

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 $\log _{10}x$ or simply as $\log x$.

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

Logarithms are commonly abbreviated as logs.

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

$c^{log_{b}a}$