Name
1.04.16
Math 4 notes and problem set
logarithms (3.3-3.4) page 1
Logarithm meanings
Objective: Understand and use several equivalent meanings of logarithms.
We’ve defined logb(c) as being the solution to the equation bx = c. There are several equivalent
ways to think about this definition:






logb(c) can be read as a question: “b to what power equals c?”
Equations can be changed back-and-forth between exp. and log. forms: bx = c  x = logb(c).
log ( c )
logb(bx) = x and b b  c .
The functions f(x) = bx and g(x) = logb(x) are inverses of each other.
The graphs of f(x) = bx and g(x) = logb(x) are related by exchanging x and y coordinates.
The graphs of f(x) = bx and g(x) = logb(x) are reflections of each other across diagonal line y = x.
Problems
1. Evaluate these logarithms without using a calculator. Hint: Read each log as a question.
a. log3 (81)
b. log3 ( 811 )
c. log 1 (81)
d. log 1 ( 811 )
3
e. log3 (–81)
3
f. log3(3k)
2. Find the value of N that would make each of these logarithms equal to 4.
a. log2(N)
b. log16(N)
c. logN(16)
d. logN(256)
e. log 1 ( N )
f. logN(a4)
2
Name
1.04.16
Math 4 notes and problem set
logarithms (3.3-3.4) page 2
For the next problem, you’ll need to know the following notations:


log10(c) may be abbreviated as log(c).
loge(c) may be abbreviated as ln(c).
(e stands for the important irrational number e ≈ 2.71828, and “ln” is read “natural log.”)
3. Evaluate these logarithms without using a calculator.
(Where the letter b appears, it stands for any positive number other than 1.)
a. log5(125)
b. logb (b5)
c. log 1 ( 19 )
3
d. log5 ( 3 5 )
e. log36 (6)
f. log2 ( 321 )
g. log13 (13)
h. log14 ( 141 )
i. logb ( b1 )
log 1 7
k. logb (bn)
l. logb( b )
m. log(10000)
n. log(0.000001)
o. log(1)
p. ln(e6)
q. ln(e)
r. ln ( 1e )
j.
7
4. Change each of these equations to the other form (either exponential or logarithmic).
17  2 =
a. log2( 641 ) = –6
b.
c. 3k = 47
d. logr(5) = t
1
49
Name
1.04.16
Math 4 notes and problem set
logarithms (3.3-3.4) page 3
5. Change each of these equations to the other form (either exponential or logarithmic).
You may need to use some algebraic steps other than the bx = c  x = logb(c) step.
Part a is completed as an example.
a. 2 ∙ 3x = 7
b. 2 + log4(x) = 17
3x = 3.5
log3(3.5) = x
c. 2000 ∙ (1.0234)x = 3000
d. log2  3x  = 5
e. a ∙ (1.06)5 = 2000
d. 9 logw(7) = n
6. For each of the given functions, write a formula for its inverse function, then on a separate
sheet of paper, sketch graphs of the pair of functions. (You may use your calculator to help
with the graphs.)
a. f(x) =
 14 x
b. f(x) = log2(x)
Answers
1. a. 4, b. –4, c. –4, d. 4, e. undefined, f. k
2. a. 16, b. 65536, c. 2, d. 4, e.
3. a. 3, b. 5, c. 2, d. 13 , e.
4. a. 2–6 =
1
64
1
2
1
16
, f. a
, f. –5, g. 1, h. –1, i. –1, j. –1, k. n, l. 12 , m. 4, n. –6, o. 0, p. 6, q. 1, r. –1
, b. log 1  491   2 , c. log3(47) = k, d. rt = 5
7
n
15
5
5. b. 4 = x, c. log1.0234(1.5) = x, d. 2 =
6. a. f –1(x) = log 1 ( x) , b. f –1(x) = 2x
4
x
3
or x = 3 ∙ 2 , e. log
5
2000
1.06
a
(
) = 5, f. w 9  7