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COLLEGE ALGEBRA MATH 110 |
Southwestern Adventist University |
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| Distance Education | Lawrence E. Turner, Jr., Ph.D. | |
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Logarithms
We have previously defined the exponential function:
f(x) = bx, b > 0, b ≠ 1
For b > 1, this is a rapidly increasing function of x as x gets large.
The domain and range of f(x) are Df = (−∞,+∞) and Rf = (0,+∞)
Previously, we have also explored the concept of inverse functions. If:
y = f(x)
and f(x) is one-to-one on a given domain, then the inverse function, f−1(x), is a function that does:
x = f−1(y)
that is, the inverse function "undoes" or "reverses" the effect of the function. Where the function changes a value, x, into another value, y, the inverse function takes this value, y, and gives back the original value, x.
The graph in the xy-plane of the inverse function is a reflection across the diagonal y = x line of the graph of the function.
The domain of the inverse function is the range of the function: Df−1 = Rf.
And the range of the inverse function is the domain of the function: Rf−1 = Df.
For polynomial functions we had a procedure for finding the inverse functions. These turned out to be polynomial functions. For a function with a radical, the inverse involved raising to a power. The exponential function is one-to-one so it should have an inverse function, but what function could it be?
We do know some things about the inverse of the exponential function. We know what its graph looks like. We know what its domain and range are, but what is it?
The domain and range of inverse of the exponential function are D = (0,+∞) and R = (−∞,+∞)
As it turns out, the inverse function of the exponential function is something new and cannot be expressed in terms of any of the functions we have previously studied. We can "invent" a new function! Actually, this was done a long time ago. The inverse function of the exponential function is the logarithmic function:
| if f(x) = bx, then f−1(x) = logb x, b > 0, b ≠ 1 |
The equation y = logb x is equivalent to x = by.
The logarithm (or log) of a number is the exponent to which the base must be raised to produce the number.
The base could be in (0,1) or (0,∞). For b on the interval (0,1), the exponential function is a decreasing function and the logaritmic function is also decreasing. On the interval (1,∞) both are increasing functions. Whereas the exponential is a rapidly increasing function for large x, the logarithmic function is a very slowly increasing function.
While the relationships and properties we will concentrate on will work for any acceptable b, the most useful values of b are a few that are greater than 1.
| base | logarithmic function | special designation | name |
| 2 | log2 x | lg x | binary logarithm |
| e | loge x | ln x | natural logarithm |
| 10 | log10 x | log x | common logarithm |
where e is the special irrational number: e ≈ 2.71828 18284 59
The logarithmic function is an extremly slowly increasing function when x gets large. The larger the base, the more slowly growing the logarithmic function becomes. Let us illustrate this with common logs; that is, base 10:
| x | log x |
| 1 | 0 |
| 10 | 1 |
| 100 | 2 |
| 1000 | 3 |
| 10,000 | 4 |
| 100,000 | 5 |
| 1,000,000 | 6 |
To get a "feel" for this, let us draw a graph of the logarithmic function on a chalk board with 1 inch for each unit on both the x and y axes, then 1 inch from the origin, the graph of the logarithmic function would cross the x-axis. At 10 inches to the right from the origin the graph would be up 1 inch. You would need to move to 100 inches (or over 8 feet) for it to get up to 2 inches. To get up to 3 inches would be at the 1000 inch mark, or over 83 feet. A 4 inch height would find us at 10,000 inches which is 833 feet. You would need to continue to the 100,000 inch (8333 feet or almost 1.6 miles) to arrive at a point that is 5 inches up.. And, the graph would need to be about 15.8 miles wide just to be 6 inches high. This is an extremly slowly growing function!
What other properties does the logarithmic function have? Are these useful? Or, are they only fun for a mathematician?
First, because we have defined the logarithmic function as the inverse function to the exponential (and the exponential function is the inverse function to the logarithmic function) we can use this property to manipulate equations. That is, an equation involving an exponential function can be changed into one involving a logarithmic function (and vice versa). As an example:
8 = 23
By taking the log2 of both sides we get:
log2 8 = log2 (23) = 3
or
3 = log2 8
While logb bx may not look like a typical composition of functions, f−1(f(x)), it really is. The reason it may look different is that we normally write the exponential function using a superscript and we normally write the logarithmic function with more that one letter and typically both without parentheses. Just don't be confused!
We can also go the other way:
4 = log5 625
By exponentiating both sides with respect to 5, we get:
54 = 5log5 625 = 625
or
625 = 54
To understand the logarithmic function it is helpful and necessary to clearly state the basic relationships that make it useful.
The first thing we can note, it that the logarithmic function has two very special values:
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These special values are true regardless of the value of b as long as it is selected from the previously established set of values: (0,1) ∪ (1,∞).
Except for some other special values, we need a calculator (or in olden days, a log table) to get approximate numerical values for the logarithm of a number. Note, the logarithmic function generally produces irrational results just like the square root function—yes, the radical is another function that is written in a "non-standard" functional notation to take a number and change it into another number!
We can also note that the y-axis is a vertical asymptote. This is true because the x-axis is a horizontal asymptote for the exponential function. The logarithmic function does not have a horizontal asymptote. In addition, the logarithm of a negative or zero value does not exist (as a real number!).
We have already discussed the relationship between the logarithmic and exponential functions:
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Note that for relationship 4. we need x > 0 so that it is within the domain of the logarithmic function.
Other logarithmic function relationships are more interesting. For two positive real numbers x and y and any real number p we have:
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Historically these are very important. Once upon a time when calculations had to be done by hand with a pencil and paper the use of logarithms made things considerably easier. It is much faster to add and subtract two values than it is to multiply them or (even worse) divide them. Relationships 5. and 6. change multiplication into addition and division into subtraction! In order to accomplish this without a calculator, we need a table of logarithms (generally for base 10—common logs).
Of course, we really want, as an example, xy and not just logb xy. However, we can exponentiate both sides of the equation using base b; that is,
xy = blog xy = b(log x + log y)
In reality, we can use the same table of logarithms to perform this last step and actually get a value for xy.
To compute 5.267817 times 6.314205 we look up the two logarithms, log (5.267817) = 0.721631 and log (6.314205) = 0.800319. These are added to obtain: 1.521949. The last step is to use the log tables backwards to find a number which has a log of 1.521949. This is: 33.262076.
The idea is that it is less effort to perform two direct lookups, one addition, and one reverse lookup than it is to perform a single multiplication. The process was the basis of the slide rule which performed multiplication and division by sliding one logarithmic scale against another to accomplish adding or subtracting lengths.
Please note that relationship 7. is consistent with relation 5. (and, for that matter, relaton 6.). As an example, let us suppose we have:
logb x2
we should be able to use any of the relations; that is, realizing the x2 = x•x and using 5. we would get:
logb x2 = logb x•x = logb x + logb x = 2 logb x
Applying relationship 7. directly gives:
logb x2 = 2 logb x
which is the same!
We could also use relation 7. to raise any number (as long as it is positive) to any power. As an example, what is 1.04524? This is a common calculation involved in computing compound interest. Clearly multiplying 1.045 times itself 23 times is not painless. However, using relation 7. we first look up the log (common log) of 1.045 to obtain 0.019116. Multiplying this by 24 gives: 0.458791. Performing a "reverse" lookup, the number that this logarithm corresponds to is 2.876014. Typically, the powers are relatively simple so that the multiplication is relatively fast and we do not need to apply relation 5.!
Of course, with a calculator all the arithmetic operations are so easy and fast! Log tables for computation have gone the way of the log cabin and the pony express!
It is important to remember that the relationships for the logarithmic functions come from the relationships for the exponential function. As an example, the exponential relation:
bx•by = b(x+y)
is directly associated with logarithmic function relation 5.
The primary useful of the logarithmic function and the relationships is to allow us to manipulate equations that contain these for certain applications in nature and finances.
As an example, consider:
2 log2 3 + log2 5
Can this be simplified? First we note that the base of both log functions are the same. If they were not, then we would probably be stuck!
Next, we see if this looks like any of the relationships above for the logarithmic function. It looks almost like relation 5. in that the expression involves the sum of two log functions. However, for relation 5. there is no multiplier coefficient for either log function. Not a problem! We can use relation 7. to "move" the coefficient of 2 "inside" the log function as an exponent:
= log2 32 + log2 5
now we can combine the two logs using relation 5. to get:
= log2 32•5
and finally we perform the arithmetic for our final result:
= log2 45
Sometimes things can even simplify further. As an example:
5 log2 8
This simplifies because the value "inside" the log function, 8, is an exact power of the base of the logarithm, 2, namely 23. We can write this as:
= 5 log2 23
using the inverse function relationship, 3. this becomes:
= 5•3 = 15
We also have one more relationship that is useful in manipulating equations:
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This last relationship can be demonstrated by exponentiating with respect to b both sides of the equation with the logs and using the inverse relationship (4. above).
A logarithm function can have any positive (≠ 1) base. Calculators typically implement the two most used ones: common (base 10) and natural (base e). So what about other bases? It turns out that there is a simple expression that permits changing from one base to another:
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Note that the two logarithmic functions on the right-hand-side are both to the same base. This expression allows us to compute a logarithm to any base. Most scientific calculators will implement both natural and common logarithms. We can use either to compute the value of the logarithm to any base such as 3.
As an example, using the natural log key:
| log3 14 = | ln 14 | ≈ | 2.63905732962 | ≈ 2.40217 |
| ln 3 | 1.09861228867 |
We could also use common logs:
| log3 14 = | log 14 | ≈ | 1.14612803568 | ≈ 2.40217 |
| log 3 | 0.47712125472 |
Expression 9. shows that because the denominator is a constant for a given base, every logarithmic function is simply a constant times any other one. The graph of logarithmic functions for different bases will all have the same shape and all pass through (1,0). The logarithm functions with larger bases will simply grow more slowly.
Finally, there are a few common errors:
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The problem with these is that look almost like the correct relationships given above, but not quite! As an example, the relationship 5. above deals with the logarithm of a product whereas, error 1. is the product of logarithms. Error 2. and 3. both deal with the ratio of logs—avoid the temptation to turn this into a difference of logs or a log of a ratio.
© 1999–2009, 2010 by Lawrence Turner