Download 9-2 Algebra 2B Logarithms and Logarithmic Functions 2017

9-2
Algebra 2B 2017: Logarithms and Logarithmic Functions
In general, the inverse of y = b x is x = b y. In x = b y, y is called the logarithm
of x. It is usually written as y = log b x and is read y equals log base b of x.
New Vocabulary
logarithm
logarithmic function
logarithmic equation
logarithmic inequality
Write each equation in logarithmic form.
1.
53
125
2.
70
1
41 1
81
4. 34 5. 3
f(x) = b x
3.
34
81
1
64
g[f(x)] = x
f (log b x) = x
g(b ) = x
x
6. 7776 6
b log b = x
Write each equation in exponential form.
7. log6 216 3
1
81
8. log2 64 6
9. log3 4
1
2
10. log10 0.00001 5
3
5
11. log25 5 12. log32 8 Evaluate each expression.
13. log9 1
1
3
17. log3 1
49
14. log8 4
1
256
18. log4 15. log7 16. log6 64
19. log9 9(n 1)
20. 2log2 32
Solve each equation or inequality. Check your solutions.
21. log15 x 3
22. log7 q 0
24. logy 16 4
25. logn 3
1
8
g(x) = log b x
f [g(x)] = x
1
5
inverses
23. log6 (2y 8) 2
26. logb 1024 5
x
log b b x = x
Proper ties of Logarithms
Suppose m and n are positive numbers, b is a positive number other
than 1, and p is any real number. Then the following properties hold.
Property
Definition
Example
Product
logb mn logb m logb n
log3 9x log3 9 log3 x
Quotient
m
logb logb m logb n
n
log14 log1 4 log1 5
Power
logb m p p logb m
log2 8x x log2 8
Equality
If logb m logb n, then m n.
log8 (3x 4) log8 (5x 2)
so, 3x 4 5x 2
4
5
4
4
27. log8 (3x 7) log8 (7x 4) 28. log7 (8x 20) log7 (x 6) 29. log3 (x2 2) log3 x
30. INVESTING Cody invests $1000 in a savings account that pays 4% interest
compounded annually. The value of the account A at the end of five years can be
determined from the equation log A log[1000(1 0.04)5]. Find the value of A to the
nearest dollar.
31. FISH The population of silver carp has
been growing in the Mississippi River.
About every 3 years, the population
doubles. Write logarithmic expression
that gives the number of years it will
take for the population to increase by
a factor of ten.
33. LOGARITHMS Ariel knows that
logb x 3 and logb y 5. She knows that
this is the same as knowing that b3 x
and b5 y. Multiply these two equations
together and rewrite it as an equation
involving logarithms. What is logb xy?
32. POWERS Jordan tries to solve the
equation log4 2x 5. She got the wrong
answer. What was her mistake? What
should the correct answer be?
34. DIGITS Jennifer, a programmer, wants
to write a formula that tells how many
digits there are in a number n, where
n is a positive integer. For example, if
n 343, the formula should evaluate to
3 and if n 10,000, the formula should
evaluate to 5. Suppose 8 log10 n 9.
How many digits does n have?
1.
log4 2x 5
2.
2x 45
3.
x 25
4.
x 32