Ex 2.3 Exponential Equations And Inequalities

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2.3 The relationship between a and logax Exponential function is y = ax. Logarithmic function is y = logax. The graphical relationship between y = ax and y = logax is the reflection in the line y = x.

The gradient of y = ex is ex. The gradient of y = ln x is x -1. The functions ex and ln x are inverse functions, the graph of y = ex and y = ln x are mirror images in the line y = x. The range of f : x  e x for x  R is f( x)  R  . The range of f : x  ln x for x  R  is f( x)  R .

2.4

Exponential Equations and inequalities

Properties of Indices: (1) (2) (3) (4)

2a × 2b = 2a + b 2a ÷ 2b = 2a – b 2a × 3a = (2×3)a = 6a (2a)b = 2a× b

Example 6: Without using table or calculator, solve the equations: (b) 2 × 4x + 1 = 1612x (c) 4x – 9(2x) + 8 = 0 (d) ex – e-x = 0 (a) 4x ×32x = 216 Example 7: Given y = axb and y = 2 when x = 3, y  Example 8: Solve the following inequalities: (a) 2x < 16 (b) 4n > 750

2 when x = 9, find a and b. 9 (c) (0.6)n < 0.2

Example 9: Find the smallest value of n for which the nth term of the geometric progression with first term 2 and common ratio 0.9 is less than 0.1. Example 10: How many terms of the geometric series 1 + 2 + 4 + 8 + … must be taken for the sum to exceed 1011?

2 Exercise 2.3: Solving exponential equations and inequalities 1. Without using table or calculator, solve the following equations: (a) 2x × 5x = 1000 (h) 4 x  32 x  6 2x x (b) 3 × 4 = 36 (i) .53 x  25 x 1  1 2 125 3 y 3  9 2 y (c) (d) (e) (f) (g) 2.

2x + 2 – 3 = 5 × 2x - 1 32x + 1 – 28(3x) + 9 = 0 42x – 68 (4x) + 256 = 0 32x + 1 – 82(3x) + 27 = 0

(j) (k) (l)

.. 3 x  3 x  2  90 5  2x  2  4x  2 2 2 x 3  2 x 3  1  2 x

The curve y = abx passes through (1, 96), (2, 1152) and (3, p). Find the values of a, b and p.

3. The curve y = axn passes through (2, 9) and (3, 4). Calculate the values of a and n. 4. Given that y = axb – 5, and that y = 7 when x = 2, and y = 22 when x = 3, find the values of a and b. 5. Solve the simultaneous equations: 3 x  9 2 y  27 1 8 6. Solve the following inequalities: (a) 5x ≥ 125 (b) 0.4n > 0.45 (c) 2x > 128 (d) 3x ≤ 243 (e) 7x ≤ 49-1 2x  4y 

7. Find the smallest possible integer n such that (a) 2n > 104 (b) 2n > 50 (c) 5.2n > 1000 (d) 0.5n < 10- 4 8. Find the largest possible integer n such that (a) 2.7n < 850 (b) 6.2n < 4000 (c) 4.6n < 30000 9. Find the least number of terms in the Geometric progression, 2 + 2.4 + 2.88 + … such that the sum exceeds 1 million. 10. How many terms of geometric series 2 + 6 + 18 + 54 + … must be taken for the sum to exceed 3 million? 11. A biological culture contains 500 000 bacteria at 12 noon on Monday. The culture increases by 10% every hour. At what time will the culture exceed 4 million bacteria?