6 of the 100, worked through step by step — try them before opening the answer.
1 log₂(16) + log₃(27) = ?
Hint: Remember that logarithms ask 'to what power must the base be raised to get the argument?' For example, log₄(64) equals 3 because 4³ = 64.
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Answer: 7
- Understand the problem We need to compute: log₂(16) + log₃(27)
- Evaluate log₂(16) We ask: "2 raised to what power equals 16?" 2^1 = 2 2^2 = 4 2^3 = 8 2^4 = 16 So, log₂(16) = 4
- Evaluate log₃(27) We ask: "3 raised to what power equals 27?" 3^1 = 3 3^2 = 9 3^3 = 27 So, log₃(27) = 3
- Add the results log₂(16) + log₃(27) = 4 + 3 = 7
- Final answer The sum is 7.
2 Compare f(x)=8x+7, g(x)=x²+9, h(x)=5^x for large x. Which function grows fastest?
Hint: Consider how each function behaves as x increases - linear functions grow steadily, quadratic functions grow faster than linear, and exponential functions eventually outpace both.
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Answer: h(x)=5^x
- Analyze f(x)=8x+7 (linear function) As x increases, f(x) grows at a constant rate of 8 per unit increase in x.
- Analyze g(x)=x²+9 (quadratic function) As x increases, g(x) grows proportionally to x squared, which is faster than linear growth.
- Analyze h(x)=5^x (exponential function) As x increases, h(x) grows by multiplying by 5 for each unit increase in x. This is exponential growth.
- Compare growth rates For small x, linear or quadratic might appear faster, but for large x: - Linear: f(x) ~ 8x - Quadratic: g(x) ~ x² - Exponential: h(x) ~ 5^x Since exponential functions (with base > 1) always grow faster than polynomial functions for sufficiently large x, h(x)=5^x grows fastest.
The answer is h(x)=5^x.
3 Compare f(x)=7x+12, g(x)=x²+7, h(x)=2^x for large x. Which function grows fastest?
Hint: Think about how each function type behaves as x becomes very large. Linear functions increase by a constant amount each step, quadratic functions increase by an amount that grows with x, and exponential functions multiply by a constant factor each step. Which type eventually overtakes the others?
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Answer: h(x)=2^x
- Identify the function types. f(x)=7x+12 is linear (constant slope 7). g(x)=x²+7 is quadratic (grows like x²). h(x)=2^x is exponential (base 2 > 1).
- Compare linear and quadratic. For large x, x² grows faster than 7x. For example, at x=100: f(100)=7(100)+12=712, g(100)=100²+7=10007. So g(x) > f(x) for large x.
- Compare quadratic and exponential. For large x, exponential functions with base > 1 eventually outgrow any polynomial. At x=10: g(10)=10²+7=107, h(10)=2¹⁰=1024. At x=20: g(20)=400+7=407, h(20)=2²⁰=1,048,576. The exponential quickly surpasses the quadratic.
- Conclusion. For sufficiently large x, h(x)=2^x grows fastest among the three functions.
The answer is h(x)=2^x.
4 Compare f(x)=6x+10, g(x)=x²+4, h(x)=2^x for large x. Which function grows fastest?
Hint: Think about how each type of function behaves as x becomes very large. Linear functions increase at a constant rate, quadratic functions increase at a rate proportional to x, and exponential functions increase by a constant factor each step.
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Answer: h(x)=2^x
- Identify the function types. f(x)=6x+10 is linear, g(x)=x²+4 is quadratic, h(x)=2^x is exponential.
- Compare linear and quadratic. For large x, the quadratic term x² dominates the linear term 6x, so g(x) grows faster than f(x).
- Compare quadratic and exponential. For large x, exponential functions with base greater than 1 (like 2^x) eventually outgrow any polynomial function (like x²). This is because exponential growth multiplies, while polynomial growth adds.
- Conclusion. For sufficiently large x, h(x)=2^x grows fastest.
The answer is h(x)=2^x.
5 Compare f(x)=11x+6, g(x)=x²+16, h(x)=6^x for large x. Which function grows fastest?
Hint: Think about how each type of function behaves as x becomes very large. Linear functions increase by a constant amount each step, quadratic functions increase by an amount that grows with x, and exponential functions multiply by a constant factor each step. Which type eventually outpaces all polynomials?
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Answer: h(x)=6^x
- Analyze f(x)=11x+6 (linear function). As x increases, f(x) grows at a constant rate of 11 per unit increase in x. For large x, f(x) ≈ 11x.
- Analyze g(x)=x²+16 (quadratic function). As x increases, g(x) grows proportionally to x squared. For large x, g(x) ≈ x². Quadratic growth is faster than linear growth because x² eventually exceeds 11x for sufficiently large x.
- Analyze h(x)=6^x (exponential function with base 6 > 1). As x increases, h(x) multiplies by 6 for each unit increase in x. For large x, h(x) ≈ 6^x.
- Compare growth rates for large x. Exponential functions with base > 1 always grow faster than any polynomial function (linear, quadratic, cubic, etc.) for sufficiently large x. Since 6^x grows much faster than x² and 11x, h(x) will eventually be the largest. Therefore, h(x)=6^x grows fastest for large x.
6 Compare f(x)=6x+11, g(x)=x²+16, h(x)=2^x for large x. Which function grows fastest?
Hint: Think about how each type of function behaves as x becomes very large. Linear functions increase at a constant rate, quadratic functions increase at a rate proportional to x, and exponential functions increase at a rate proportional to their current value.
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Answer: h(x)=2^x
- Analyze f(x)=6x+11 (linear function). As x increases, f(x) grows by a constant 6 for each unit increase in x. For large x, f(x) behaves like 6x.
- Analyze g(x)=x²+16 (quadratic function). As x increases, g(x) grows proportionally to x². For large x, the +16 becomes negligible, so g(x) behaves like x².
- Analyze h(x)=2^x (exponential function). As x increases, h(x) doubles for each unit increase in x. For large x, this growth far exceeds any polynomial growth.
- Compare growth rates for large x. For any exponential function with base greater than 1, it will eventually outgrow any polynomial function. Since 2^x grows faster than x², and x² grows faster than 6x, the order from slowest to fastest is: f(x) (linear), g(x) (quadratic), h(x) (exponential). Therefore, h(x)=2^x grows fastest for large x.