CH 3: Calculating the period of an investment
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Questions and Answers

What is the general form of an exponential function?

  • f(x) = x^b
  • f(x) = b + x
  • f(x) = b * x
  • f(x) = b^x (correct)

    • What condition must be satisfied for the base b in an exponential function?

  • b > 0 and b ≠ 1 (correct)
  • b > 0 and b = 1
  • b = 0 and b ≠ 1
  • b < 0 and b ≠ 1

    • What happens when the base b in an exponential function is greater than 1?

  • The function exhibits exponential decay.
  • The function exhibits exponential growth. (correct)
  • The function oscillates.
  • The function remains constant.

    • What is the y-intercept of the graph of an exponential function f(x) = b^x?

    <p>(0, 1)</p> Signup and view all the answers

    What is the domain of an exponential function?

    <p>All real numbers</p> Signup and view all the answers

    What is the range of an exponential function?

    <p>All positive real numbers</p> Signup and view all the answers

    What is the horizontal asymptote of an exponential function?

    <p>y = 0</p> Signup and view all the answers

    What is the general form of a logarithmic function?

    <p>y = log_b(x)</p> Signup and view all the answers

    What condition must be satisfied for the base b in a logarithmic function?

    <p>b &gt; 0 and b ≠ 1</p> Signup and view all the answers

    If f(x) = log_b(x), what is the value of f(b)?

    <p>0</p> Signup and view all the answers

    What is the inverse operation of the logarithm function?

    <p>Exponential function</p> Signup and view all the answers

    What is the domain of the logarithm function?

    <p>Positive real numbers</p> Signup and view all the answers

    What is the range of the logarithm function?

    <p>All real numbers</p> Signup and view all the answers

    What is the intercept of the logarithm function?

    <p>(1, 0)</p> Signup and view all the answers

    What is the asymptotic behavior of the logarithm function?

    <p>Vertical asymptote at x = 0</p> Signup and view all the answers

    What is the rule for multiplying two numbers using logarithms?

    <p>Product Rule: $\log_b(xy) = \log_b(x) + \log_b(y)$</p> Signup and view all the answers

    What is the rule for dividing one number by another using logarithms?

    <p>Quotient Rule: $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$</p> Signup and view all the answers

    What is the rule for exponentiation using logarithms?

    <p>Power Rule: $\log_b(x^p) = p \cdot \log_b(x)$</p> Signup and view all the answers

    What is the formula used to convert the base of a logarithm?

    <p>Change of Base Formula: $\log_b(x) = \frac{\log_k(x)}{\log_k(b)}$</p> Signup and view all the answers

    Which of the following statements about solving exponential equations is true?

    <p>It requires isolating the exponential term and applying logarithmic operations to both sides.</p> Signup and view all the answers

    What is the main difference between exponential growth and exponential decay?

    <p>Exponential growth occurs when the base b is greater than 1, while exponential decay occurs when the base b is between 0 and 1.</p> Signup and view all the answers

    What is the range of the logarithmic function $y = \log_b(x)$?

    <p>The range is all positive real numbers (0, $\infty$).</p> Signup and view all the answers

    What is the value of $\log_b(b)$?

    <p>1</p> Signup and view all the answers

    Which of the following statements about the domain of an exponential function $f(x) = b^x$ is correct?

    <p>The domain is all real numbers, i.e., $(-\infty, \infty)$.</p> Signup and view all the answers

    What is the rule for converting the base of a logarithm from b to a?

    <p>$\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$</p> Signup and view all the answers

    What is the horizontal asymptote of the exponential function $f(x) = b^x$?

    <p>The horizontal asymptote is the line $y = 0$.</p> Signup and view all the answers

    Which of the following statements about the y-intercept of the graph of the exponential function $f(x) = b^x$ is correct?

    <p>The y-intercept is always 1, regardless of the base b.</p> Signup and view all the answers

    What is the inverse operation of the logarithm function?

    <p>The inverse operation of the logarithm function is the exponential function.</p> Signup and view all the answers

    What is the rule for exponentiation using logarithms?

    <p>$b^{\log_b(x)} = x$</p> Signup and view all the answers

    What happens when the base b in an exponential function $f(x) = b^x$ is greater than 1?

    <p>The function exhibits exponential growth.</p> Signup and view all the answers

    If $\log_2(x) = 5$, what is the value of $x$?

    <p>$2^5$</p> Signup and view all the answers

    Given $\log_3(x) + \log_3(y) = 2$, what is the value of $\log_3(xy)$?

    <p>2</p> Signup and view all the answers

    If $\log_5(x) = 2$ and $\log_5(y) = -1$, what is the value of $\log_5\left(\frac{x}{y}\right)$?

    <p>3</p> Signup and view all the answers

    If $\log_2(x) = 3$, what is the value of $\log_4(x)$?

    <p>$\frac{3}{\log_2(4)}$</p> Signup and view all the answers

    If $\log_2(x) = 4$, what is the value of $\log_2(x^3)$?

    <p>12</p> Signup and view all the answers

    If $f(x) = \log_2(x)$, what is the value of $f(16)$?

    <p>4</p> Signup and view all the answers

    If $\log_3(x) = 2$ and $\log_3(y) = 4$, what is the value of $\log_3\left(\frac{x^2}{y}\right)$?

    <p>-2</p> Signup and view all the answers

    If $\log_2(x) = 3$ and $\log_4(y) = 2$, what is the value of $\log_8(xy)$?

    <p>$\frac{11}{3}$</p> Signup and view all the answers

    If $\log_5(x) = 2$ and $\log_5(y) = -1$, what is the value of $\log_5\left(\frac{x^2}{y^3}\right)$?

    <p>5</p> Signup and view all the answers

    If $\log_3(x) = 2$ and $\log_3(y) = -1$, what is the value of $\log_9\left(\frac{x^2}{y}\right)$?

    <p>$\frac{5}{2}$</p> Signup and view all the answers

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