Week 4 - Part 1 - Properties of Logarithms and Logarithmic Functions

When you have finished this lesson and worked your way through Chapter 4 Section 4.1 of the text you should be able to:

1. Know the definition of a logarithm
2. Convert between a logarithmic function and an exponential function
3. Use the basic properties of logarithms to simplify logarithmic expressions or equations
4. Understand the difference between natural and common logarithms
5. 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= log₂x. 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 = 2x		x = 2y
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 -- 10² = 100
• common logarithm -- Log 1 = 0, or as a power -- 10⁰ = 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 -- e⁰ = 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 propoerties hold for both common (log) and natural (ln) logarithms.

Properties of Common Logarithms
1. log(A) only exists for A > 0 (The argument of the log must be positive and greater than zero.)
2. log(A*B) = log(A) + log(B)
3. log(A÷B) = log(A) - log(B)
4. log(A^p) = p log(A)
5. log(10^x) = x for all x
6. log(1) = 0
7. log(10) = 1

Properties of Natural Logarithms
1. ln(A) only exists for A > 0 (The argument of the ln must be positive and greater than zero.)
2. ln(A*B) = ln(A) + ln(B)
3. ln(A÷B) = ln(A) - ln(B)
4. ln(A^p) = p ln(A)
5. ln(e^x) = x for all x
6. ln(1) = 0
7. 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(x³) = 3 log(x)
• log(1÷x²) = log(x^-2) = -2 log(x) (Because 1/X² = 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(x²y³) = log(x²) + log(y³)

  = 2 log(x) + 3 log(y)

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

1. log(ab)
2. ln[(ab)/c]
3. log(ab³)
4. ln(a + b)
5. log[(a+b)/c]
6. log(10x³z²)

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Answers to questions

1. log(ab) = log a + log b
2. ln[(ab)/c] = ln(ab) - ln c = ln a + ln b - ln c
3. log(ab³) = log a + log b³ = log a + 3 log b
4. ln(a + b) - cannot be further simplified (a + b) is NOT THE SAME as (ab) multiplied
5. log[(a+b)/c] = log (a+b) - log c
6. log(10x³z²) = log 10 + log x³ + log z² = 1 + 3 log x + 2 log z

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