# Exponentials & Logarithms

Contents

### Summary

• $f\left( x \right) \quad =\quad { a }^{ x }$ is known as an exponential function
• Logarithmic functions are written as $\log _{ a }{ b\quad =\quad x\quad \Leftrightarrow \quad { a }^{ x } } \quad =\quad b$
• Laws of logarithm:

1.     $\log _{ a }{ x } \quad +\quad \log _{ a }{ y } \quad =\quad \log _{ a }{ (xy) }$

2.     $\log _{ a }{ x } \quad -\quad \log _{ a }{ y } \quad =\quad \log _{ a }{ (\frac { x }{ y } ) }$

3.     $\log _{ a }{ { x }^{ k } } \quad =\quad k\log _{ a }{ x }$

4.     $\log _{ a }{ { x }^{ -1 } } \quad =\quad -\log _{ a }{ x }$

5.     $\log { { (10) }^{ x } } \quad =\quad x$

6.     ${ 10 }^{ (log\quad x) }\quad =\quad x$

#### Exponential functions

When n is a positive integer and a is the base in ${ a }^{ n }$, then n is known as the index, power or exponent.

Suppose we have $f\left( x \right)\quad =\quad { a }^{ x }$, this is called an exponential function where a is a positive constant and power x is a variable. You are quite familiar with an exponential function ${ e }^{ x }$, here also $e\quad =\quad 2.718$,  which is a positive constant. Examples of other exponential functions include ${ 2 }^{ x },\quad { 50 }^{ x },\quad { 10 }^{ x }$ etc.

Graph of an exponential function is always increasing i.e see Fig 1

Some important points to remember for exponential functions are:

1.     f(x) is always positive.

2.     When x = 0, f(x) = 1

3.     It is always an increasing function.

4.     When x decreases towards $-\infty$, it approaches zero but never reaches zero.

#### Logarithms

Like all functions, exponential functions have inverses. The inverse of the exponential is the logarithm, or log, for short.

The logarithmic functions are written as $\log _{ a }{ b } \quad =\quad x$ which means the same as ${ a }^{ x }\quad =\quad b$.

In $\log _{ a }{ b } \quad =\quad x$ a is called the base, logs can have different bases, however the most common one is base 10. The symbol “log” on calculators also means log with base 10.

To help you understand log notation better lets just look at a few examples.

#### Examples

1.     ${ 10 }^{ 2 }\quad =\quad 100\quad \Leftrightarrow \quad \log _{ 10 }{ 100\quad =\quad 2 }$

2.     ${ 5 }^{ 3 }\quad =\quad 125\quad \Leftrightarrow \quad \log _{ 5 }{ 125\quad =\quad 3 }$

3.     ${ 100 }^{ 2 }\quad =\quad 10.000\quad \Leftrightarrow \quad \log _{ 100 }{ 10.000\quad =\quad 2 }$

#### Law2s of Logarithms

Some important laws of logarithm that you must remember are:

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

• Subtracting Logs

$\log _{ a }{ x } \quad -\quad \log _{ a }{ y } \quad =\quad \log _{ a }{ (\frac { x }{ y } ) }$

• Power of logs

$\log _{ a }{ { x }^{ k } } \quad =\quad k\log _{ a }{ x }$

$\log _{ a }{ { x }^{ -1 } } \quad =\quad -\log _{ a }{ x }$

$\log { { (10) }^{ x } } \quad =\quad x$

${ 10 }^{ (log\quad x) }\quad =\quad x$

Note: We can’t take logs of negative numbers

#### Example #1

Q. Simplify      $\log _{ 10 }{ 2 } \quad +\quad \log _{ 10 }{ 8 }$

Solution:

Using the law of logarithm we know:   $\log _{ a }{ x } \quad +\quad \log _{ a }{ y } \quad =\quad \log _{ a }{ (xy) }$

Hence:

$\log _{ 10 }{ 2 } \quad +\quad \log _{ 10 }{ 8 } \quad =\quad \log _{ 10 }{ (2\quad x\quad 8) }$

Ans:              $=\quad \log _{ 10 }{ (16) }$

Another important thing about logarithm is Change of Base.

We should be able to change the base of a log as many questions require this i.e

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

#### Example #2

Q. Solve    $\log _{ 7 }{ 4 }$

Solution:

We know a = 7, x = 4. It will be easier to solve the question by changing base 7 to base 10 as it can be easily solved on the calculator.

Use the formula above:

$\log _{ 7 }{ 4 } \quad =\quad \frac { \log _{ 10 }{ 4 } }{ \log _{ 10 }{ 7 } }$

Ans:              $=\quad 0.7124$

#### Example #3

Suppose we have an equation   ${ (6) }^{ x }\quad =\quad 22$

We solve it by taking logarithm of both sides.

When we take logarithm of the left hand side, it becomes $x\log _{ 10 }{ 6 }$ hence the equation becomes:

$x\log _{ 10 }{ 6 } \quad =\quad \log _{ 10 }{ 22 }$

So $x\quad =\quad \frac { \log _{ 10 }{ 22 } }{ \log _{ 10 }{ 6 } }$,  we then solve this on the calculator.

Natural Log

There is an exception for e = 2.718…, we don’t use the normal log for this constant.

We use natural log which is “ln”. Natural log is just normal log with base e and it has its own symbol on the calculator known as “ln”.

Laws for natural log are the same as normal log:

$\ln { x } \quad +\quad \ln { y } \quad =\quad \ln { (xy) }$

$\ln { x } \quad -\quad \ln { y } \quad =\quad \ln { (\frac { x }{ y } ) }$

$\ln { ({ x }^{ k }) } \quad =\quad k\ln { (x) }$

$\ln { ({ e }^{ x }) } \quad =\quad x$

${ e }^{ (\ln { x } ) }\quad =\quad x$

#### Example #4

Q. Solve the equation     $\log _{ 3 }{ (2\quad -\quad 3x) } \quad -\quad 2\log _{ 3 }{ (x) } \quad =\quad 2$

Solution:

First simplify the expression with the law     $\log _{ a }{ { x }^{ k } } \quad =\quad k\log _{ a }{ x }$

$\log _{ 3 }{ (2\quad -\quad 3x) } \quad -\quad \log _{ 3 }{ ({ x }^{ 2 }) } \quad =\quad 2$

Then we use the law of subtraction:    $\log _{ a }{ x } \quad -\quad \log _{ a }{ y } \quad =\quad \log _{ a }{ (\frac { x }{ y } ) }$

$\log _{ 3 }{ (2\quad -\quad 3x) } \quad -\quad \log _{ 3 }{ ({ x }^{ 2 }) } \quad =\quad 2$

$\log _{ 3 }{ (\frac { 2\quad -\quad 3x }{ { x }^{ 2 } } ) } \quad =\quad 2$

Now take exponential of both sides:

$\frac { 2\quad -\quad 3x }{ { x }^{ 2 } } \quad =\quad { 3 }^{ 2 }\quad =\quad 9$

$2\quad -\quad 3x\quad =\quad 9{ x }^{ 2 }$

$9{ x }^{ 2 }\quad +\quad 3x\quad -\quad 2\quad =\quad 0$

$(3x\quad -\quad 1)(3x\quad +\quad 2)$

$x\quad =\quad \frac { 1 }{ 3 } ,\quad \frac { -2 }{ 3 }$

Ignore the negative solution:

Ans:              so,  $x\quad =\quad \frac { 1 }{ 3 }$

##### Reference
1. http://caps.unm.edu/mathrefresh/assets/LogsandExp.pdf
2. https://math.stackexchange.com/questions/90594/the-difference-between-log-and-ln
3. CGP AS-level mathematics edexcel revision and practice book