Properties of Logarithms
Here are the properties that hold for logarithms:
1) log b m + log b n = log b mn
2) log b m − log b n = log b
3) x log b m = log b (m x )
4) loga b =
where b > 0 b ≠1, m, n >0
m
n
where b > 0 b ≠1, m, n >0
where b > 0 b ≠1, m >0
log m b
log m a
Problems:
1) Simplify or express as a single logarithm:
a) ln e4 b) log 4 − log 3 + log π + 3log r 2) On the same set of axes graph y = 3x and y = log 3 x
c) 101+ 2 log x
3) Graph y = log x .
4) Solve:
a) log 4 (2x + 1) − log 4 (x − 2) = 1 b)
log 2 (x 2 + 8) = log 2 x + log 2 6
5) Express y in terms of x:
b)
1
ln y = (ln 4 + ln x)
3
c) ln y – ln x = 2 ln 7
d) log 3 ( x ) = log 3 ( y ) − 2
f) log ( yx ) = 9
a) log y = 2 log x
e) ln
1
= −5x y
6) Write each expression as a single log of a single argument.
a) log 5 12 − log 5 3 c) log 2 225 − log 2 5 + log 2 3 e) 4 log b (x 2 y) + 3 log b (y 2 ) + log b 2 b) ln(6) + ln(5) + ln(4)
d) 3 log 4 5 − log 4 9
f) ln(16) – ln(2x) + 3lnx – 2ln2
7) If a = log(9), b = log(35), and c = log(2), rewrite log(2520) in terms of a, b, and c.
8) Solve the following by taking the log (base 10) of each side.
a) 2x = 3 9) If f(x) = k log 5 x , where k is constant, then f(5) – f(25) = ?
b) e 2x = 25 c) 3(0.4)x = 8
10) a5
32
If log 2 b = log 3
and log 3 a = log 2
, find the values of a and b.
81
b
11) Given log 5 = a, log 3 = b, and 3x +2 = 45, solve for x in terms of a and b.
12) If log 2 2 a =
2
a
(a > 0), solve for a.
4
13) Circle all the following that are correct:
a)
d)
b)
e)
c)
f)
g)
h)
j)
k)
m)
n)
p)
q)
t)
u)
w)
x)
i)
l)
o)
r)
s)
v)
14) Given log 2 = a , log 3 = b , and log 7 = c , find log(1i2i3i4i5i6i7i8i9i10) in terms of a,
b, and c.
15) If xy = 8 , find the value of 3log 2 x + 3log 2 y .
16) Let f (x) = 3x . Find f −1 (1) .
17) State the domain and range of y = log 5
(
3
)
x−6 .
⎛ b⎞
18) Given: log 3 ⎜ ⎟ = log 5 5a 2 and log 3 b = log 5 ( 625a ) , find the values of a and b.
⎝ 9⎠
( )
19) Evaluate. Remember, do NOT use a calculator.
⎛ 45 8 ⎞
a) log 2 ⎜
⎟ ⎝ 2 ⎠
c) 34 log 3 5 d)
b) log 2 56 − log 8 343
log 7 16
1
log 7
16
20) Express as one logarithm: a) 21)
( )
(
)
2 log xy 3 − 3log x 2 y 2 + log
(
)
x 2 y6 b)
⎛ 1⎞
1
2
log1/b y 2 + log b y 4 − log b ⎜ ⎟
⎝ y⎠
6
3
( )
( )
Evaluate (you may use a calculator).
a) log 5 45 b) log 2 0.45 22) Given log 2 3 = a and log 9 7 = b , find log 2
c) 3 log0.4 3
7
in terms of a and b.
2
23) Find all solutions for x. Be sure to check your answer:
3log9 (x +1) = x − 1
24) Write as a single logarithm, or number, with no coefficient before the log:
( )
a) 4 loga x 5 y − loga x 2 (
)(
b) log x y 4 log y x 4
)
(
)
c) ( log x y ) log y x e)
d) log b y 6 + 2 log b x1/3 − log
log 5 x log 3 x10
+
log 5 a 2 log 3 a 4
25) Use these values to find the values for the logarithms given below.
log b k = 3.25
log b m = −0.233
log b n = 1.08
a) log b k 3 n −5 ⎛ k ⎞
b) log b ⎜
⎝ mn ⎟⎠
c) log k b d) ( log k m ) log b k 2
(
26) Give the following logarithm values:
log b a = 0.25193 log b c = 1.43068 a) Find logc b ⎛ 1⎞
b) Find log b ⎜ 2 ⎟ ⎝d ⎠
)
log b d = −1
⎛ ac ⎞
c) Find log b ⎜ ⎟
⎝ d⎠
b
y
28) Given a = log 3 and b = log 5:
⎛ 2⎞
a) Find log ⎜ ⎟ in terms of a and b.
⎝ 3⎠
b) Find log 6 72 in terms of a and b.
29) Given log 36 2 = a , find each of the following in terms of a:
a) log 36 3 b) log 36 18 c) log 36 12
d) log 36 96 e) log 36
f) log 3 12
g) log 2 12 h) log 3 2 i) log 9 8
b) (log 3 x)2 + log 3 x 2 + 1 = 0
9
8
30) Solve for x.
a) log12 (log 9 (log 5 (log 2 32))) = x