When you have finished this lesson (and the material in your textbook) you should be able to:

- Know the definition of a logarithm
- Convert between a logarithmic function and an exponential function
- Use the basic properties of logarithms to simplify logarithmic expressions or equations
- Understand the difference between natural and common logarithms
- Graph and manipulate logarithmic functions

**The Relationship Between Exponential and Logarithmic Functions**

Consider an exponential function of the form y = ab^{x}. "b" is usually referred to as the base and x as the exponent. If we substitute values for x we can quickly calculate values for y. For example let y = 2^{x}. Y is equal to 2 to the power x. We could perform similar operations with other bases (3, 4, 10, etc...) The first two columns of the table given below show the values of y for various values of x. As you can see from the table increasingly negative values of x lead to smaller and smaller values of y. The curve approaches the x axis asymptotically. This means it gets closer and closer to the x axis but doesn't cross it. On the other hand as x becomes positive and increases the value of y also increases.

If we now interchange x and y the equation we are looking at becomes x = a^{y}. Unfortunately we have not yet covered the techniques for solving this equation for y. So, let's **plug-in values of y and calculate the resulting values of x.** The last two columns of the table show the value of x for various values of y. There are several things that you should note from the data shown. First when y is negative x is positive but decreasing in value. As a matter of fact no matter how negative y becomes x is always positive, although it does get smaller and smaller.

The figure shown to the right of the table depicts a portion of the graph of each function and the line y = x. The line y = x is included because the two equations are reflections of each other across this line. Thus the point (0,1) on the curve y =2^{x} is reflected as the point (1,0) on the curve x = 2^{x}. These curves are inverses of one another. In mathematics you write the second equation as follows y= log_{2}x. In common language this says y is the log of x to the base 2. What it means is that y is equal to 2 to the power x (y = 2^{x}).

y = 2^{x} | x = 2^{y} |
|||
---|---|---|---|---|

Value of x | Value of y | ------- | Value of y | Value of x |

3 | 8 | 3 | 8 | |

2 | 4 | 2 | 4 | |

1 | 2 | 1 | 2 | |

0 | 1 | 0 | 1 | |

-1 | 1/2 | -1 | 1/2 | |

-2 | 1/4 | -2 | 1/4 | |

-3 | 1/8 | -3 | 1/8 |

If you look at your calculator you are likely to see keys labeled "log" and "ln". By common agreement log means base 10 and ln means base e (2.71.......). Logs to base 10 are called common logarithms and logs to base e (ln) are called natural logarithms. So y = log(x) means x = 10^{y} while y = ln(x) means x = e^{y}.
A few examples are probably in order at this point.

- common logarithm -- Log 100 = 2, or as a power -- 10
^{2}= 100- common logarithm -- Log 1 = 0, or as a power -- 10
^{0}= 1- common logarithm -- Log 0.1 = -1 or as a power -- 10
^{-1}= 1/10 or 0.1

Similarly

- natural logarithm -- ln 100 = 4.6052, or as a power -- e
^{4.6052}= 100- natural logarithm -- ln 1 = 0, or as a power -- e
^{0}= 1- natural logarithm -- ln 0.1 = -2.3026, or as a power -- e
^{-2.3026}= 0.1 (due to rounding 0.099999)

We will discuss a number of basic properties of logarithms. These properties hold for both common (log) and natural (ln) logarithms.

Properties of Common Logarithms

- log(A) only exists for A > 0 (The argument of the log must be positive and greater than zero.)
- log(A*B) = log(A) + log(B)
- log(A÷B) = log(A) - log(B)
- log(A
^{p}) = p log(A)- log(10
^{x}) = x for all x- log(1) = 0
- log(10) = 1

Properties of Natural Logarithms

- ln(A) only exists for A > 0 (The argument of the ln must be positive and greater than zero.)
- ln(A*B) = ln(A) + ln(B)
- ln(A÷B) = ln(A) - ln(B)
- ln(A
^{p}) = p ln(A)- ln(e
^{x}) = x for all x- ln(1) = 0
- ln(e) = 1

Some examples

Arguments Multiplied

- log(3x) = log(3) + log(x)
- log(xyz) = log(x) + log(y) + log(z)
- log(3 + x) = cannot be simplified, arguments (3 and x) are added not multiplied
Arguments Divided

- log(x/y) = log(x) - log(y)
- log(x/8) = log(x) - log(8)
- log(x/10) = log(x) - log(10) = log(x) - 1 (because log(10) = 1)
Arguments to a Power

- log(x
^{3}) = 3 log(x)- log(1÷x
^{2}) = log(x^{-2}) = -2 log(x) (because 1/X^{2}= X^{-2})Putting it all together

- log[4(1+x)
^{3}÷(1-x)] = log(4) + log(1+x)^{3}- log(1-x)= log(4) + 3 log(1+x) - log(1-x)- log(x
^{2}y^{3}) = log(x^{2}) + log(y^{3})= 2 log(x) + 3 log(y)

Practice Questions - Expand the following using the logarithmic properties given above:

- log(ab)
- ln[(ab)/c]
- log(ab
^{3})- ln(a + b)
- log[(a+b)/c]
- log(10x
^{3}z^{2})

Answers to questions

- log(ab) = log a + log b
- ln[(ab)/c] = ln(ab) - ln c = ln a + ln b - ln c
- log(ab
^{3}) = log a + log b^{3}= log a + 3 log b- ln(a + b) - cannot be further simplified (a + b) is NOT THE SAME as (ab) multiplied
- log[(a+b)/c] = log (a+b) - log c
- log(10x
^{3}z^{2}) = log 10 + log x^{3}+ log z^{2}= 1 + 3 log x + 2 log z