Exponential And Logarithmic Functions! Math Trivia Quiz

1. y = 2^x is a(n)_____.

Exponential function

Logarithmic function

Power function

Polynomial function

The equation y = 2x represents an exponential function because it follows the form y = ab^x, where a is the initial value and b is the base. In this case, the base is 2, and as x increases, y also increases exponentially.

Explanation

The equation y = 2x represents an exponential function because it follows the form y = ab^x, where a is the initial value and b is the base. In this case, the base is 2, and as x increases, y also increases exponentially.

2. For the graph of y = b^x, the domain is _____.

All positive real numbers

All real numbers

All negative real numbers

All natural numbers

The graph of y = bx represents a straight line that extends infinitely in both the positive and negative directions. This means that for any real number x, there will be a corresponding value of y. Therefore, the domain of the graph is all real numbers.

Explanation

The graph of y = bx represents a straight line that extends infinitely in both the positive and negative directions. This means that for any real number x, there will be a corresponding value of y. Therefore, the domain of the graph is all real numbers.

3. Solve the equation log₅ (3x + 9) = log₅ (5x - 3).

3

4

5

6

The equation log5 (3x + 9) = log5 (5x - 3) implies that the logarithms of the two expressions are equal. Since the base of the logarithm is the same (5), we can set the two expressions inside the logarithms equal to each other. Therefore, 3x + 9 = 5x - 3. By rearranging the equation, we get 2x = 12, and solving for x gives x = 6.

Explanation

The equation log5 (3x + 9) = log5 (5x - 3) implies that the logarithms of the two expressions are equal. Since the base of the logarithm is the same (5), we can set the two expressions inside the logarithms equal to each other. Therefore, 3x + 9 = 5x - 3. By rearranging the equation, we get 2x = 12, and solving for x gives x = 6.

4. Find the value of log₉219 using the change of base formula.

0.2601

1.3862

2.4527

3.1021

The change of base formula allows us to find the value of a logarithm with a base other than 10 or e. In this case, we can use the formula log(base a) of b = log(base c) of b / log(base c) of a. By applying this formula, we can find the value of log(base 9) of 219 by dividing the logarithm of 219 with base 10 by the logarithm of 9 with base 10. The result is approximately 2.4527.

Explanation

The change of base formula allows us to find the value of a logarithm with a base other than 10 or e. In this case, we can use the formula log(base a) of b = log(base c) of b / log(base c) of a. By applying this formula, we can find the value of log(base 9) of 219 by dividing the logarithm of 219 with base 10 by the logarithm of 9 with base 10. The result is approximately 2.4527.

5. Given that log 5 = 0.6990, evaluate log 5000.

5.6990

4.6990

3.6990

2.6990

The logarithm of a number represents the exponent to which a base must be raised to obtain that number. In this case, we are given that log 5 = 0.6990. To evaluate log 5000, we need to find the exponent to which the base 5 must be raised to obtain 5000. Since 5^4 = 625, and 5^5 = 3125, we can conclude that 5000 is between these two values. Therefore, the exponent must be between 4 and 5. Since the answer choices are in the form of 0.6990, we can deduce that the correct answer is 3.6990, as it falls between 3 and 4.

Explanation

The logarithm of a number represents the exponent to which a base must be raised to obtain that number. In this case, we are given that log 5 = 0.6990. To evaluate log 5000, we need to find the exponent to which the base 5 must be raised to obtain 5000. Since 5^4 = 625, and 5^5 = 3125, we can conclude that 5000 is between these two values. Therefore, the exponent must be between 4 and 5. Since the answer choices are in the form of 0.6990, we can deduce that the correct answer is 3.6990, as it falls between 3 and 4.

6. Suppose the population in a city can be modeled by the equation
y = 15,000(1.02)^x, where x is the number of years since 1960. Predict the population in 2020.

49,215

523,333

33,967

5,486

The equation given models the population of a city over time. The equation shows exponential growth, with the base of the exponent being 1.02. This means that the population is increasing by 2% each year. To predict the population in 2020, we need to find the value of x for that year. Since 2020 is 60 years after 1960, we can substitute x = 60 into the equation. Evaluating the equation, we get y = 15,000(1.02)^60 ≈ 49,215. Therefore, the predicted population in 2020 is 49,215.

Explanation

The equation given models the population of a city over time. The equation shows exponential growth, with the base of the exponent being 1.02. This means that the population is increasing by 2% each year. To predict the population in 2020, we need to find the value of x for that year. Since 2020 is 60 years after 1960, we can substitute x = 60 into the equation. Evaluating the equation, we get y = 15,000(1.02)^60 ≈ 49,215. Therefore, the predicted population in 2020 is 49,215.

7. Determine the amount of money in a money market account providing an annual rate of 6.5% compounded daily if the invested amount of $3,500 is left in the account for 5 years.

$3,735.03

$3,943.84

$1,343.97

$4,843.97

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Explanation

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8. What is the y-intercept of the graph of y = 2^x + 5?

(0, 6)

(0, 1)

(0, 5)

(0, 32)

The y-intercept of a graph represents the point where the graph intersects the y-axis. In the equation y = 2x + 5, the y-intercept is the value of y when x is 0. Plugging in x = 0 into the equation, we get y = 2(0) + 5 = 5. Therefore, the y-intercept is (0, 5).

Explanation

The y-intercept of a graph represents the point where the graph intersects the y-axis. In the equation y = 2x + 5, the y-intercept is the value of y when x is 0. Plugging in x = 0 into the equation, we get y = 2(0) + 5 = 5. Therefore, the y-intercept is (0, 5).

9. Determine the amount of money in a money market account providing an annual rate of 7% compounded daily if Philips invested $2500 and left it in the account for 10 years.

$4917.88

$4974.47

$5034.04

$4915.25

The correct answer is $5034.04. To determine the amount of money in the money market account, we can use the formula for compound interest: A = P(1 + r/n)^(nt), where A is the final amount, P is the principal amount, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years. Plugging in the given values, we have A = 2500(1 + 0.07/365)^(365*10) = $5034.04.

Explanation

The correct answer is $5034.04. To determine the amount of money in the money market account, we can use the formula for compound interest: A = P(1 + r/n)^(nt), where A is the final amount, P is the principal amount, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years. Plugging in the given values, we have A = 2500(1 + 0.07/365)^(365*10) = $5034.04.

10. Solve the equation 3^4x = 8^{x - 5} by using natural logarithms.

-4.4912

-4.3150

-0.6583

-0.2859

To solve the equation 34x = 8x - 5 using natural logarithms, we can start by isolating the variable x. Subtracting 8x from both sides gives us 26x = -5. Next, we can divide both sides by 26 to solve for x. Taking the natural logarithm of both sides allows us to simplify the equation further. The natural logarithm of e raised to the power of x is simply x. Applying this property, we can rewrite the equation as ln(e^(26x)) = ln(e^(-5)). Simplifying further, we get 26x = -5. Taking the natural logarithm of both sides again, we have ln(26x) = ln(-5). Solving for x using a calculator, we find that x is approximately -4.4912.

Explanation

To solve the equation 34x = 8x - 5 using natural logarithms, we can start by isolating the variable x. Subtracting 8x from both sides gives us 26x = -5. Next, we can divide both sides by 26 to solve for x. Taking the natural logarithm of both sides allows us to simplify the equation further. The natural logarithm of e raised to the power of x is simply x. Applying this property, we can rewrite the equation as ln(e^(26x)) = ln(e^(-5)). Simplifying further, we get 26x = -5. Taking the natural logarithm of both sides again, we have ln(26x) = ln(-5). Solving for x using a calculator, we find that x is approximately -4.4912.