1. Exponential Equations

2. Do you remember how to solve 2 x = 16?  2 x = 16 An equation with unknown indices is called an exponential equation. 2 x = 2 4  x = 4 Example of an Exponential equation Express both sides as powers of 2

3. How to solve 2 2x + 2 x  6 = 0? Since 2 2x = (2 x ) 2, we can transform the original equation into a quadratic equation in 2 x.

4. Let me try to solve the equation. 2 2x + 2 x  6 = 0 (2 x ) 2 + 2 x  6 = 0 Let y = 2 x, the equation becomes y 2 + y  6 = 0 (y  2)(y + 3) = 0 y = 2 or y =  3  2 x = 2 or 2 x =  3 (rejected) 2 x = 2 1  x = 1 2 x is positive.

5. Follow-up question 2 2x + 4(2 x )  32 = 0 Let y = 2 x, the equation becomes y 2 + 4y  32 = 0 (y  4)(y + 8) = 0 y = 4 or y =  8 2 x = 2 2  x = 2  2 x = 4 or 2 x =  8 (rejected) Solve 2 2x + 4(2 x )  32 = 0. (2 x ) 2 + 4(2 x )  32 = 0

6. How to solve the following simultaneous equations? By writing 81 = 3 4 and 4 = 2 2, the simultaneous exponential equations can be simplified. 3 x + y = 81 ……(1) 2  x + 5y = 4 ……(2)

7. Let me try to solve the equations. From (1), 3 x + y = 3 4 From (2), 2  x + 5y = 2 2 3 x + y = 81 ……(1) 2  x + 5y = 4 ……(2) x + y = 4  x + 5y = 2 x + y = 4 ……(3)  x + 5y = 2 ……(4) The simultaneous equations become: Solving (3) and (4), we have

8. Follow-up question From (1), 2 x + 2y = 2 0 2 x + 2y = 1 ……(1) 5 x + 3y = ……(2) x + 2y = 0 ……(3) x + 3y =  1 ……(4) The simultaneous equations become: Solving (3) and (4), we have Solve the following simultaneous equations: x + 2y = 0 From (2), 5 x + 3y = 5  1 x + 3y =  1