1.6 Exponentials and Logarithms

• What is the general form of an exponential function?

  $y =$$a^{x}$
• The definition of a logarithmic function involves finding the exponent to which a base must be raised.


  True
• What happens to a decreasing exponential function as x approaches positive infinity?

  Approaches 0
• Exponential functions can either increase or decrease depending on the base value.


  True
• What is the general form of a logarithmic function?

  $x =$$\log_{a} y$
• The negative exponent rule states that $a^{ - x} =$$1 / a^{x}$, which means a negative exponent results in a reciprocal fraction
• The product rule of logarithms states that $\log_{a} (x \cdot y) =$$\log_{a} x +$$\log_{a} y$, which simplifies the logarithm of a product into the sum of two logarithms
• In the exponential form y = a^x</latex>, $a$ is the base and $x$ is the exponent
• Increasing exponential functions grow faster as x increases.


  True
• Exponentials and logarithms are inverse mathematical operations that relate to powers and bases.
• The definition of an exponential function involves raising a base to an exponent.


  True
• Match the property of exponential functions with its description:

  Base ↔️ a > 0

  Increasing Behavior ↔️ Increases as x increases

  Asymptotic Behavior ↔️ Approaches infinity as x grows
• What happens to an increasing exponential function as x approaches positive infinity?

  Approaches positive infinity
• Logarithms are used to determine the exponent to which a base must be raised.
• Logarithmic functions are the inverse of exponential functions.


  True
• What type of asymptote do logarithmic functions have?

  Horizontal
• The laws of exponents allow exponential expressions to be simplified and combined.


  True
• What is the change of base formula for logarithms?

  $\log_{b} x =$$\frac{\log_{a} x}{\log_{a} b}$
• What does $\log_{2} 8$ equal?


  3
• \(\log_a y = x\) means that \(y\) equals \(a\) raised to the power of x
• What is the general form of a logarithmic function?

  x = log_a y
• A decreasing exponential function approaches 0 as \(x\) becomes large.


  True
• The general form of an exponential function is \(y = a^x\), where \(a\) is the base
• If \(a > 1\), the exponential function \(y = a^x\) increases as \(x\) increases
• Logarithmic functions are the inverse of exponential
• A logarithmic function has a horizontal asymptote at \(y = 0\) as \(x\) approaches infinity
• A negative exponent means taking the reciprocal
• To convert \(y = a^x\) to logarithmic form, it becomes \(x = \log_a\) y
• Converting between exponential and logarithmic forms is based on their inverse relationship.


  True
• It is necessary to check solutions to exponential equations for validity.


  True
• The change of base formula allows converting logarithms to a more convenient base.

  True
• What is the general form of an exponential equation?

  \(y = a^x\)
• Match the property with the type of exponential function:

  Increasing as x increases ↔️ Base a > 1

  Decreasing as x increases ↔️ Base 0 < a < 1
• What determines whether an exponential function is increasing or decreasing?

  The base `a`
• A decreasing exponential function approaches positive infinity as `x` gets small
• Logarithmic functions have a slower rate of growth compared to exponential functions.
• Any number raised to the power of 0 equals 1.


  True
• Simplify log_4 16^3 using the power rule of logarithms.

  6
• Convert \(2^3 = 8\) to logarithmic form.

  \(\log_2 8 = 3\)
• The logarithmic form of \(y = a^x\) is x