Logarithm and Exponential Questions with Answers and Solutions - Grade 12

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

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

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3. Use log rule of product: log₄(10) = log₄(2) + log₄(5)
   log₄(2) = log₄(4^1/2) = 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

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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

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

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6. Use change of base formula: (log_xa)(log_ab)
   = log_xa (log_xb / log_xa) = log_xb

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7. 2 = log_ae
   a² = e
   ln(a²) = ln e
   2 ln a = 1
   a = e^1/2

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

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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

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10. Note that b^{4 log_bx} = x⁴
    The given equation may be written as: 2x x⁴ = 486
    x = 243^1/5 = 3

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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.

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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

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13. Given: 9^x - 3^x - 8 = 0
    Note that: 9^x = (3^x)²
    Equation may be written as: (3^x)² - 3^x - 8 = 0
    Let y = 3^x and rewite equation with y: y² - 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
    3^x = ( 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
    

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14. Given: 4^{x - 2} = 3^{x + 4}
    Take ln of both sides: ln ( 4^{x - 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 (3⁴ * 4²) / ln (4/3) = ln (3⁴ * 2⁴) / ln (4/3)
    = 4 ln(6) / ln(4/3)
    

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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

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