MATH 135: PRECALCULUS
DEFINITION & PROPERTIES OF LOGARITHMS
1. Properties of logarithms. Graph each of the following logarithmic functions.
(Label three points to be sure you have the shape right.) Give the domain and
range of each.
i. f (x) = log2 (x)
4
Domain: (0, ∞)
2
Range: (−∞, ∞)
−2
(2, 1)
1 (0, 1) 2
( 12 , −1)
3
4
−4
8
ii. g(x) = 3 − ln(x − 1)
Domain: (1, ∞)
6
(1 + 1e , 4)
4
(2, 3)
(1 + e, 2)
Range: (−∞, ∞)
2
1
2
3
4
2. Separating logarithms. Write each of the following expressions as a sum or
difference of logarithms of a single factor.
i. ln
x2 − 9
3
ii. log2
= ln(x2 − 9) − ln 3 = ln(x − 3) + ln(x + 3) − ln 3
x2
√
3
x+1
√
1
= log2 (x2 ) − log2 ( 3 x + 1) = 2 log2 (x) − log2 (x + 1)
3
3. Combining logarithms. Write each of the following expressions as a single
logarithm.
i. log5 (2) + 3 log5 (x) = log5 (2) + log5 (x3 ) = log5 (2x3 )
ii.
1
2
ln(x + 1) − 7 ln(x − 2) = ln
√
7
x + 1 − ln (x − 2)
√
= ln
x+1
(x − 2)7
4. Changing base. Write each of the following logarithms in terms of the natural
logarithm ln(x). If you brought your calculator, approximate the number to two
decimal places.
i. log2 (5) =
ln(5)
≈ 2.32
ln(2)
ii. log7 (3) =
ln(7)
≈ 1.77
ln(2)