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Worksheet 3.3A, Exponents and Logarithms
MATH 1410
(SOLUTIONS)
1. f (x) = log2 (x). Fill in the table for the values of x and f (x) and then graph y = f (x) using the
points you have plotted.
x
16
1/4
1/64
32
1/32
1/4
1
2
8
1/1024
0
f (x)
4
−4
−6
5
−5
−2
0
1
3
−10
DNE
2. Write the following statements in exponential form. Then solve for x.
(a) log2 (32) = x
(b) log2 (217 ) = x
(c) log2 ( 18 ) = x
√
(d) log2 ( 3 2) = x
√
(e) log2 ( 2) = x
(f) ln(e10 ) = x
(g) log2 (43 ) = x
(h) log2 (430 ) = x
Solutions.
(a) log2 (32) = x =⇒ 2x = 32 so x = 5.
(b) log2 (217 ) = x =⇒ 2x = 217 so x = 17.
(c) log2 ( 18 ) = x =⇒ 2x = 18 so x = −3
√
√
(d) log2 ( 3 2) = x =⇒ 2x = 3 x so x = 31 .
√
√
(e) log2 ( 2) = x =⇒ 2x = 2 so x = 12 .
(f) ln(e10 ) = x =⇒ ex = 310 so x = 10.
(g) log2 (43 ) = x =⇒ 2x = 43 = (22 )3 = 26 so x = 6.
(h) log2 (430 ) = x =⇒ 2x = 430 = (22 )30 = 260 so x = 60.
3. Draw the graph of each of the following functions and give the domain and range. (Think in terms
of transformations.)
(a) y = log2 (x)
(b) y = log2 (x + 1)
(c) y = log2 (x − 1)
(d) y = log2 (2x)
(e) y = log2 (x) + 1
(f) y = 2 log2 (x)
(g) y = log2 (x2 )
Solution.
(a) y = log2 (x) has domain: (0, ∞) and range: (−∞, ∞).
(b) y = log2 (x + 1) shifts the first graph one unit to the left and so has domain: (−1, ∞) and
range: (−∞, ∞).
(c) y = log2 (x − 1) shifts the first graph one unit to the right and so has domain: (1, ∞) and
range: (−∞, ∞).
(d) y = log2 (2x) shrinks the first graph horizontally by a factor of 2. This does not change the
domain or range and so this function has domain: (0, ∞) and range: (−∞, ∞).
(e) y = log2 (x) + 1 shifts the first graph up one. It has domain: (0, ∞) and range: (−∞, ∞).
(f) y = 2 log2 (x) stretches the first graph vertically by a factor of 2. This does not change the
domain or range and so the domain is (0, ∞) and the range is (−∞, ∞).
(g) y = log2 (x2 ) has domain (−∞, 0) ∪ (0, ∞) since we can now input both positive and negative
numbers for x. The range is still (−∞, ∞).
4. Suppose log10 (2) = a and log10 3 = b. Use properties of exponents to write out the following
logarithms. (In most problems your answers will involve a and b.)
(a) log10 (4).
(b) log10 (6).
(c) log10 (8).
(d) log10 (9).
(e) log10 (24).
1
(f) log10 ( ).
2
(g) log10 (10).
(h) log10 (5).
Solutions.
(a) log10 (4) = 2 log 2 = 2a.
(b) log10 (6) = log 2 + log 3 = a + b.
(c) log10 (8) = 3 log 2 = 3a.
(d) log10 (9) = 2 log 3 = 2b.
(e) log10 (24) = 3 log 2 + log 3 = 3a + b.
1
(f) log10 ( ) = − log 2 = −a.
2
(g) log10 (10) = 1
(h) log10 (5) = log 10 − log 2 = 1 − a.
5. Find the inverse function of f (x).
(a) f (x) = 10x .
(b) f (x) = ln(x)
2
(c) f (x) = ex .
(d) f (x) = ex
2
−5
.
x
(e) f (x) = 5 + e .
(f) f (x) = log2 (x + 2) + 2.
Solution.
(a) y = log x
(b) y = ex
√
(c) y = ln x.
√
(d) y = ln x + 5
(e) y = ln(x − 5)
(f) y = 2x−2 − 2.