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Difficulty: HardLogarithmic and Exponential Expressions and Equations

If log⁡2(a)+log⁡2(b)=5\log_2(a) + \log_2(b) = 5 and log⁡2(a2)−log⁡2(b)=4\log_2(a^2) - \log_2(b) = 4 for positive real numbers aa and bb, what is the value of a+ba + b?

  1. A
    5
  2. B
    10
  3. 12Answer
  4. D
    16
  5. E
    32

Answer

12
The correct answer is 12. We can simplify the system of equations by using the power rule of logarithms, which allows us to rewrite log⁡2(a2)\log_2(a^2) as 2log⁡2(a)2\log_2(a). Letting x=log⁡2(a)x = \log_2(a) and y=log⁡2(b)y = \log_2(b) gives us the system x+y=5x + y = 5 and 2x−y=42x - y = 4. Adding these equations gives 3x=93x = 9, which means x=3x = 3. Substituting this back gives y=2y = 2. Converting back from logarithmic form to exponential form, we get a=23=8a = 2^3 = 8 and b=22=4b = 2^2 = 4. Therefore, a+b=8+4=12a + b = 8 + 4 = 12.

Step-by-Step Solution

1
Use the power property of logarithms, log⁡b(xk)=klog⁡b(x)\log_b(x^k) = k \log_b(x), to rewrite the second equation.
The equation log⁡2(a2)−log⁡2(b)=4\log_2(a^2) - \log_2(b) = 4 becomes 2log⁡2(a)−log⁡2(b)=42\log_2(a) - \log_2(b) = 4.
This simplifies the term log⁡2(a2)\log_2(a^2) so that it is linear in terms of log⁡2(a)\log_2(a).
2
Substitute variables to simplify solving the system of equations. Let x=log⁡2(a)x = \log_2(a) and y=log⁡2(b)y = \log_2(b).
The system of equations becomes:
1) x+y=5x + y = 5
2) 2x−y=42x - y = 4
Variable substitution reduces the logarithmic system to a standard system of linear equations.
3
Solve the linear system by adding the two equations together.
Adding the equations yields (x+y)+(2x−y)=5+4(x + y) + (2x - y) = 5 + 4, which simplifies to 3x=93x = 9, so x=3x = 3. Substituting x=3x = 3 back into the first equation gives 3+y=53 + y = 5, so y=2y = 2.
Addition eliminates the variable yy, allowing us to solve for xx and then find yy.
4
Convert the solved values of xx and yy back into aa and bb using the exponential form definition of a logarithm.
Since x=log⁡2(a)=3x = \log_2(a) = 3, we have a=23=8a = 2^3 = 8. Since y=log⁡2(b)=2y = \log_2(b) = 2, we have b=22=4b = 2^2 = 4.
The definition of a logarithm log⁡b(z)=w\log_b(z) = w is equivalent to z=bwz = b^w.
5
Calculate the sum of aa and bb.
a+b=8+4=12a + b = 8 + 4 = 12.
To find the final value requested by the question stem.

Key Concept

Solving systems of logarithmic equations using logarithm properties and exponential conversions
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Topics

Logarithmic and Exponential Expressions and Equations

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