# Exponential Functions andLogarithms Explained

Here we will look at exponential and logarithmic functions, what they represent and how they relate to each other in Algebra.

### Exponential Functions

If we have a function of the form    f(x) = a^x,     a > 0.

This is an exponential function.

The base number is  a,  and the exponent is  x.

If we recall:    3^2 \space {\small{=}} \space 3 \times 3 \space {\small{=}} \space 9     ,     3^3 \space {\small{=}} \space 3 \times 3 \times 3 \space {\small{=}} \space 27

### Logarithmic Functions

A logarithm is a function that goes the other way from an exponential one.

It is the inverse function to the exponential function.

Such a function has the general form.    f(x) \space {\small{=}} \space{\tt{log}}_a (x)

With   a > 0 \space \space , \space \space a \space {\cancel{=}} \space 1 \space \space , \space \space x > 0.
( We’ll explain these conditions further in the page. )

We know,     4^2 \space {\small{=}} \space 4 \times 4 \space {\small{=}} \space 16.

The base number is  4,  the argument is  x,  and the exponent is  2.

As a logarithmic function we write,    {\tt{log}}_4 (16) \space {\small{=}} \space 2.

So the result is, what does the base number  a  have to be raised to in order to get  x.

It’s the case that exponential and logarithmic functions are inverses of each other.
One undoes the work of the other.

{\tt{log}}_a (a^x) \space {\small{=}} \space x         =>         a^{{\tt{log}}_a (x)} \space {\small{=}} \space x

So if we have,   f(x) \space {\small{=}} \space{\tt{log}}_4 (x).     Then,   4^{f(x)} \space {\small{=}} \space x.

## Logarithms Explained, Base 10 and Base e

### Logarithm Base 10

A logarithm with base  10  is known as a ‘common logarithm’.     \boxed{{\tt{log}}_{10} (x)}

It’s the case that the ’10’ is generally left unwritten,
but is implied/assumed.

So when we encounter,  {\tt{log}} (x).
This is assumed to be,  {\tt{log}}_{10} (x).

{\tt{log}} (100)‘   would be assumed to be   ‘{\tt{log}}_{10} (100)‘.

{\tt{log}} (100) \space {\small{=}} \space {\tt{log}}_{10} (100) \space {\small{=}} \space 2

The ‘log’ button on a calculator is the logarithm of base  10.

### Logarithm Base e

A logarithm of base  e  is known as a ‘natural logarithm’.     \boxed{{\tt{log}}_{e} (x)}

This logarithm is often written  {\tt{ln}}(x),

{\tt{log}}_{e} (x) \space {\small{=}} \space {\tt{ln}} (x)

Like logarithms with base  10,  the natural logarithm also appears as a button on a calculator.

If we wanted to find the natural logarithm of  825.

We can use a calculator to establish.   {\tt{ln}}(825) \space {\small{=}} \space 6.715

For both the common logarithm and natural logarithm, it’s the case that:

10^{{\tt{log}}(x)} \space {\small{=}} \space x     and     e^{{\tt{ln}}(x)} \space {\small{=}} \space x.

## Logarithms, 0, 1,Negative Numbers and Decimals

Earlier in the page it was stated that for    f(x) \space {\small{=}} \space{\tt{log}}_a (x).

We needed   a > 0 \space \space , \space \space a \space {\cancel{=}} \space 1 \space \space , \space \space x > 0.

But why is this?

When we’re dealing with real numbers, certain issues occur when we don’t constrain the values of  a  and  x.

Let’s look at them.

### Base Value

Can’t have 0.

Having logarithms with base  0  would give us situations like the following.

{\tt{log}}_0 (2) \space {\small{=}} \space x       =>       0^x \space {\small{=}} \space 2
{\tt{log}}_0 (4) \space {\small{=}} \space x       =>       0^x \space {\small{=}} \space 4

Now  0  raised to any power is still  0,  so this base can’t occur for a logarithm.

Can’t have 1.

Having logarithms with base  1  would give us situations like the following.

{\tt{log}}_1 (2) \space {\small{=}} \space x         =>       1^x \space {\small{=}} \space 2
{\tt{log}}_1 (12) \space {\small{=}} \space x       =>       1^x \space {\small{=}} \space 12

Now  1  raised to a power is still  1,  so this base can’t occur for a logarithm.

Can’t be negative.

Having logarithms with negative base numbers would give us situations like the following.

{\tt{log}}_{{\text{–}}5} (x) \space {\small{=}} \space {\Large{\frac{1}{2}}}         =>       {\text{–}}5^{\frac{1}{2}} \space {\small{=}} \space x         =>       \sqrt{{\text{–}}5} \space {\small{=}} \space x

Now with real numbers we can’t take the root of a negative number, so this base couldn’t occur for a logarithm.

### Argument

Can’t be negative.

Having logarithms with negative argument numbers would give us situations like the following.

{\tt{log}}_{10} ({\text{–}}100) \space {\small{=}} \space x       =>       10^x \space {\small{=}} \space {\text{–}}100

The issue is that any real  x  will produce a positive result.

10^3 \space {\small{=}} \space 1000     ,     10^{{\text{–}}3} \space {\small{=}} \space {\Large{\frac{1}{1000}}}     ,     10^0 \space {\small{=}} \space 1       etc

Can’t be Zero.

Having logarithms with a zero argument would give us situations like the following.

{\tt{log}}_{10} (0) \space {\small{=}} \space x       =>       10^x \space {\small{=}} \space 0

There isn’t an x value that can satisfy this result.

### Outputs and Decimals

To conclude the logarithms explained page, the output of a logarithm can be a negative number just fine.

For example.

{\tt{log}}_{5} (0.2) \space {\small{=}} \space {\text{-}}1       =>       5^{{\text{–}}1} \space {\small{=}} \space {\Large{\frac{1}{5}}} \space {\small{=}} \space 0.2

We are also fine to use decimal numbers with logarithms, provided they satisfy the value constraints.

{\tt{log}}_{10} (14) \space {\small{=}} \space 1.146128…       =>       10^{1.146128…} \space {\small{=}} \space 14

{\tt{log}}_{10} (27) \space {\small{=}} \space 1.88649…       =>       10^{1.88649…} \space {\small{=}} \space 27

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