Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
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.