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Difficulty: MediumExponential Functions and Equations

If 8x2y=64\frac{8^x}{2^y} = 64, which of the following equations correctly expresses yy in terms of xx?

  1. A
    y=3x−64y = 3x - 64
  2. y=3x−6y = 3x - 6Answer
  3. C
    y=3x+6y = 3x + 6
  4. D
    y=6−3xy = 6 - 3x

Answer

y=3x−6y = 3x - 6
To solve the equation, express all terms using the common base of 2. Substituting 8=238 = 2^3 and 64=2664 = 2^6 into the equation gives (23)x2y=26\frac{(2^3)^x}{2^y} = 2^6. Applying the power rule of exponents transforms the numerator into 23x2^{3x}, which leads to the equation 23x2y=26\frac{2^{3x}}{2^y} = 2^6. Using the quotient rule of exponents, the division simplifies to 23x−y=262^{3x - y} = 2^6. Since the bases are now identical, their exponents must be equal: 3x−y=63x - y = 6. Isolating the variable yields the correct formulation.

Step-by-Step Solution

1
Rewrite all bases in the equation as powers of the common base 2.
Since 8=238 = 2^3 and 64=2664 = 2^6, the equation becomes (23)x2y=26\frac{(2^3)^x}{2^y} = 2^6.
Expressing all exponential terms using a single base allows for the application of exponent rules.
2
Apply the power of a power property to simplify the numerator on the left side.
The term (23)x(2^3)^x simplifies to 23x2^{3x}, yielding the equation 23x2y=26\frac{2^{3x}}{2^y} = 2^6.
The exponent rule (am)n=amn(a^m)^n = a^{mn} states that a base raised to a power and then another power requires multiplying the exponents.
3
Apply the quotient property of exponents to combine the left side of the equation.
The left side simplifies to 23x−y2^{3x - y}, resulting in 23x−y=262^{3x - y} = 2^6.
The quotient rule aman=am−n\frac{a^m}{a^n} = a^{m-n} states that dividing powers with the same base requires subtracting the exponent of the denominator from the exponent of the numerator.
4
Set the exponents equal to each other and solve for the desired variable.
3x−y=63x - y = 6, which simplifies to y=3x−6y = 3x - 6 when solved for yy.
Since the bases on both sides of the equation are equal, their exponents must also be equal.

Key Concept

Solving exponential equations by base conversion and rules of exponents

Alternative Method

Instead of converting both sides to base 2, you can also substitute a test value for xx. For example, if x=2x = 2, then the equation becomes 822y=64\frac{8^2}{2^y} = 64, which simplifies to 642y=64  ⟹  2y=1  ⟹  y=0\frac{64}{2^y} = 64 \implies 2^y = 1 \implies y = 0. Substituting x=2x = 2 into the correct option yields y=3(2)−6=0y = 3(2) - 6 = 0, confirming the answer.
Estimated Time:1m 30s
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Topics

Exponential Functions and Equations

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