Download 5.3: Logarithms - Lone Star College

5.3.1
5.3: Logarithms
Looking at the graph of f ( x) = b x , we can see it is one-to-one. So every exponential function has an
inverse.
Example 1:
Consider the function f ( x) = 3x .
Then f (1) = 3 , f (2) = 9 , f (3) = 27 , and f (4) = 81 . This function has an inverse.
So f −1 (3) = 1 , f −1 (9) = 2 , f −1 (27) = 3 , and f −1 (81) = 4 .
We call this inverse function a logarithmic function and denote it f −1 ( x) = log 3 x .
So f −1 (3) = log 3 3 = 1 and f −1 (9) = log 3 9 = 2 . Also log 3 27 = 3 and log 3 81 = 4 .
Every exponential function has an inverse. The inverses of exponential functions are called logarithmic
functions (logarithms or logs for short).
Definition: log b x = y means b y = x .
The functions f ( x) = b x and g ( x) = log b x are inverses of each other.
b is called the base of the logarithm.
5.3.2
Evaluating logarithms:
Example 2:
Evaluate log 2 8 .
In other words, log 2 8 is a number. What number is it?
This question is asking us to find a certain exponent. Specifically, “what exponent must I put on the 2 to
give me 8?”
Said another way, “2 raised to what power is 8?”
Examples:
log 2 32 = _______
log 5
( 5 ) = _______
4
log 5 1 = _______
⎛1⎞
log 3 ⎜ ⎟ = _______
⎝9⎠
log10 100 = _______
⎛ 1 ⎞
log10 ⎜
⎟ = _______
⎝ 1000 ⎠
log 3 3 = _______
log 1 1 = _______
2
log 3 3 = _______
log 2
( 16 ) = _______
3
⎛ 1 ⎞
log 3 ⎜
⎟ = _______
⎝ 3⎠
log 64 8 = _______
5.3.3
Evaluating more complicated logarithms:
Example 3:
log 4 32 = _______
Example 4:
⎛ 1 ⎞
log 9 ⎜ ⎟ = _______
⎝ 27 ⎠
The natural logarithm:
Remember we said e was a very important number? It is so important that the logarithmic function of
base e has its own special notation and its own button on your calculator.
The logarithm of base e is called the natural logarithm, which is abbreviated “ln”.
log e x = ln x
Example 5:
ln e 4 = _______
Example 6:
⎛1⎞
ln ⎜ 3 ⎟ = _______
⎝e ⎠
5.3.4
Example 7:
Evaluate ln e .
Example 8:
⎛ 1 ⎞
Simplify ln ⎜
⎟.
3 5
⎝ e ⎠
Example 9:
Evaluate ln1 .
Example 10: Evaluate log 2 (−4) .
Example 11: Evaluate log 5 0 .
IMPORTANT:
You cannot apply a logarithm to zero
or to a negative number!!!
Exponential and logarithmic forms for an equation:
Remember, log b x = y means b y = x .
Logarithmic form: log b x = y
Exponential form: b y = x
5.3.5
Example 12: Convert each of the following to exponential form.
a) log10 1000 = 3
b) log a 178 = w
c) log 7 ( y − 3) = x
d) log x 6 = y 2 + 2
Example 13: Convert each of the following to logarithmic form.
a) 7 x = 23
b) y x −1 = 8
c) x8 = u
d) ( x − 3) 2 = 6
about the relationship between f ( x) = b x and g ( x) = log b x :
Because f ( x) = b x and g ( x) = log b x are inverses of one another, f ( g ( x)) = x and g ( f ( x)) = x .
This gives us…
log b b x = x
b logb x = x
5.3.6
Example 14: Simplify log 3 3
x+ 5
.
Example 15: Simplify log 2 212 .
Example 16: Simplify ln e−2 .
Example 17: Simplify 5log5 y .
Example 18: Simplify 3log3
Example 19: Simplify eln( x
2
2
.
+1)
.
The common logarithm:
Often log is used to mean log10 . The logarithm of base 10 is called the common logarithm.
Example 20: Evaluate log 3 10 .
Solving simple logarithmic equations:
Example 21: Solve for x.
log 2 ( x − 1) = 5
Example 22: Solve for x.
log x 7 =
1
2
5.3.7
Properties of Logarithms
Laws (properties, rules) of logarithms:
1. log b b = 1
2. log b 1 = 0
3. log b ( PQ) = log b P + log b Q
⎛P⎞
4. log b ⎜ ⎟ = log b P − log b Q
⎝Q⎠
⎛1⎞
Note: this results in log b ⎜ ⎟ = − log b Q .
⎝Q⎠
5. log b P n = n log b P
6. blogb P = P
7. log b b P = P
log b x
(Change of base formula)
log b a
ln x
Note: this results in log a x =
.
ln a
8. log a x =
Example 1: Simplify log 2 56 − log 2 7 .
Example 2: Simplify log 6 4 + log 6 9 .
Example 3: Simplify log 50 − log 2 + log 4 .
5.3.8
10
Example 4: Simplify log 2 8 .
Example 5: Simplify 5
2 log 5 3
.
Example 6: Simplify e6ln 2 .
⎛3x⎞
Example 7: Simplify log 6 ⎜⎜
⎟⎟ .
36
⎝
⎠
Errors to avoid:
log a ( x + y ) ≠ log a x + log a y
log a x
≠ log a x − log a y
log a y
(log a x)3 ≠ 3log a x
Putting the properties together:
Example 8: Use the properties of logarithms to expand the expression as much as possible.
log 3 ( x( x + 4))
Example 9: Use the properties of logarithms to expand the expression as much as possible.
⎛ x+2⎞
ln ⎜
⎟
⎝ e ⎠
5.3.9
Example 10: Use the properties of logarithms to expand the expression as much as possible.
⎛ ab3 ⎞
ln ⎜ 2 ⎟
⎝c d ⎠
Example 11: Use the properties of logarithms to expand the expression as much as possible.
⎡ x+5 ⎤
ln ⎢ 2
⎣ x − 4 ⎥⎦
Example 12: Use the properties of logarithms to expand the expression as much as possible.
⎡
⎤
x +1
log10 ⎢ 2
⎥
⎣ x ( x − 3) x + 7 ⎦
Example 13: Use the properties of logarithms to expand the expression as much as possible.
ln
( x − 3) 2 ( x + 9) 4
x 6 ( x − 10)
5.3.10
Example 14: Rewrite as a single logarithm with a coefficient of 1.
log 3 x + log 3 2
Example 15: Rewrite as a single logarithm with a coefficient of 1.
log a b − c log a d + r log a s
Rewrite as a single logarithm with a coefficient of 1.
2 ln x − 5ln( x + 1) − 12 ln( x − 3) + ln( x + 2)
Rewrite as a single logarithm with a coefficient of 1.
1
[3log 5 ( x + 2) − 2 log 5 ( x − 1) + log 5 x − 4 log 5 ( x − 7)]
2
Example 16: Rewrite ln 60 so that it contains only logs that are being applied to prime numbers.
Example 17: Rewrite log 72 so that it contains only logs that are being applied to prime numbers.
5.3.11
Evaluating logs on your calculator:
Example 23: Evaluate log10 72 on your calculator.
Example 24: Evaluate ln12 on your calculator.
Example 18: Use a calculator to approximate log 2 17 to the nearest hundredth.
Example 19: Use a calculator to evaluate log 7 12 to the nearest thousandth.
About calculators and books:
•
Most calculators and some books use
o ln for natural log (base e).
o log for log10 .
•
Some math books (usually very advanced ones or those from other countries)
o use log for natural log.
o That’s all…logs of other bases are useless.
•
Some books (this is my preference)
o use ln for natural log.
o use log10 for log10 .
o never write log by itself—too confusing!!