Logarithm and Exponential Questions

Logarithm and Exponential Questions (with Answers and Solutions) 1. Solve the equation (1/2)2x + 1 = 1

2. Solve x ym = y x3 for m.

3. Given: log8(5) = b. Express log 4(10) in terms of b.

4. Simplify without calculator: log6(216) + [ log(42) - log(6) ] / log(49)

5. Simplify without calculator: ((3 -1 - 9-1) / 6)1/3

6. Express (logxa)(logab) as a single logarithm.

7. Find a so that the graph of y = logax passes through the point (e , 2).

8. Find constant A constant A such  such that log 3 x = A = A log  log5x for all x > 0.

9. Solve for x the equation log [ log (2 + log 2(x + 1)) ] = 0

10. Solve for x the equation 2 x b4 logbx = 486

11. Solve for x the equation ln (x - 1) + ln (2x - 1) = 2 ln (x + 1)

12. Find the x intercept of the graph of y = 2 log( sqrt(x - 1) - 2)

13. Solve for x the equation 9x - 3x - 8 = 0

14. Solve for x the equation 4x - 2 = 3x + 4

15. If logx(1 / 8) = -3 / 4, than what is x?

Solutions to the Above Problems 1. Rewrite equation as (1/2)2x + 1 = (1/2)0 Leads to 2x + 1 = 0 Solve for x: x = -1/2

2. Divide all terms by x y and rewrite equation as: ym - 1 = x2 Take ln of both sides (m - 1) ln y = 2 ln x Solve for m: m = 1 + 2 ln(x) / ln(y)

3. Use log rule of product: log 4(10) = log4(2) + log4(5) log4(2) = log4(41/2) = 1/2 Use change of base formula to write: log4(5) = log8(5) / log8(4) = b / (2/3) , since log8(4) = 2/3 log4(10) = log4(2) + log4(5) = (1 + 3b) / 2

4. log6(216) + [ log(42) - log(6) ] / log(49) = log6(63) + log(42/6) / log(72) = 3 + log(7) /2 log(7) = 3 + 1/2 = 7/2

5. ((3-1 - 9-1) / 6)1/3 = ((1/3 - 1/9) / 6) 1/3 = ((6 / 27) / 6) 1/3 = 1/3

6. Use change of base formula: (log xa)(logab) = logxa (logxb / log xa) = logxb

7. 2 = logae a2 = e ln(a2) = ln e 2 ln a = 1 a = e1/2

8. Use change of base formula using ln to rewrite the given equation as follo ws ln (x) / ln(3) = A ln(x) / ln(5) A = ln(5) / ln(3)

9. Rewrite given equation as: log [ log (2 + log2(x + 1)) ] = log (1) , since log(1) = 0. log (2 + log2(x + 1)) = 1 2 + log2(x + 1) = 10 log2(x + 1) = 8 x + 1 = 28 x = 28 - 1

10. Note that b4 logbx = x4 The given equation may be written as: 2x x 4 = 486 x = 2431/5 = 3

11. Group terms and use power rule: ln (x - 1)(2x - 1) = ln (x + 1) 2 ln function is a one to one function, hence: (x - 1)(2x - 1) = (x + 1) 2 Solve the above quadratic function: x = 0 and x = 5 Only x = 5 is a valid solution to the equation given above since x = 0 is not in the domain of the expressions making the equations.

12. Solve: 0 = 2 log( sqrt(x - 1) - 2) Divide both sides by 2: log( sqrt(x - 1) - 2) = 0 Use the fact that log(1)= 0: sqrt(x - 1) - 2 = 1 Rewrite as: sqrt(x - 1) = 3 Raise both sides to the power 2: (x - 1) = 32 x-1=9 x = 10

13. Given: 9x - 3x - 8 = 0 Note that: 9x = (3x)2 Equation may be written as: (3x)2 - 3x - 8 = 0 Let y = 3x and rewite equation with y: y 2 - y - 8 = 0 Solve for y: y = ( 1 + sqrt(33) ) / 2 and ( 1 - sqrt(33) ) / 2 Since y = 3 x, the only acceptable solution is y = ( 1 + sqrt(33) ) / 2 3x = ( 1 + sqrt(33) ) / 2 Use ln on both sides: ln 3 x = ln [ ( 1 + sqrt(33) ) / 2] Simplify and solve: x = ln [ ( 1 + sqrt(33) ) / 2] / ln 3

14. Given: 4x - 2 = 3x + 4 Take ln of both sides: ln ( 4x - 2 ) = ln ( 3 x + 4 ) Simplify: (x - 2) ln 4 = (x + 4) ln 3 Expand: x ln 4 - 2 ln 4 = x ln 3 + 4 ln 3 Group like terms: x ln 4 - x ln 3 = 4 ln 3 + 2 ln 4 Solve for x: x = ( 4 ln 3 + 2 ln 4 ) / (ln 4 - ln 3) = ln (3 4 * 42) / ln (4/3) = ln (3 4 * 24) / ln (4/3) = 4 ln(6) / ln(4/3)

15. Rewrite the given equation using exponential form: x- 3 / 4 = 1 / 8 Raise both sides of the above equation to the power -4 / 3: (x- 3 / 4)- 4 / 3 = (1 / 8) - 4 / 3 simplify: x = 8 4 / 3 = 24 = 16