2.4 Modeling scenarios with logarithmic functions

• The logarithmic function $y =$$\log_{b}(x)$ can be rewritten as $x =$$b^{y}$.


  True
• What is the condition for the base $b$ of a logarithmic function?


  $b > 0$, $b \neq 1$
• What is the value of $\log_{2}(8)$?


  $3$
• Match the scenario with the appropriate variable:

  pH calculation ↔️ pH

  Richter scale ↔️ Magnitude of earthquake

  Sound intensity ↔️ Decibels
• What parameter is used in the Richter scale to calculate the magnitude of an earthquake?

  Amplitude of seismic waves
• The domain of a logarithmic function is x > 0
• The base of a logarithmic function must satisfy the condition b > 0
• The Richter Scale uses logarithmic functions to calculate the magnitude of earthquakes.


  True
• Match the scenario with its independent and dependent variables:

  Richter Scale ↔️ Amplitude of seismic waves ||| Magnitude of earthquake

  Sound Intensity ↔️ Intensity of sound waves ||| Decibels
• What is the inverse of a logarithmic function?

  Exponential function
• What is the vertical asymptote of a logarithmic function?

  $x =$$0$
• What is the horizontal asymptote of an exponential function?

  $y =$$0$
• What type of relationship is modeled by logarithmic functions?

  Decreasing rate
• In scenarios such as the Richter scale, the logarithmic function provides a more convenient and intuitive representation of the data
• What is the logarithmic function used in pH calculation?

  $pH =$$- \log_{10}([H⁺])$
• What is the general form of a logarithmic regression model?

  y = a + b \log(x)</latex>
• Linear regression models a straight line relationship between variables.
• Logarithmic regression assumes constant variance in the data.


  True
• What does the correlation coefficient $r$ measure in regression?


  Strength and direction of relationship
• What is the domain of a logarithmic function?

  $x > 0$
• Match the scenario with its application of logarithmic functions:

  pH Calculation ↔️ Determining acidity of a solution

  Richter Scale ↔️ Measuring earthquake magnitude

  Sound Intensity ↔️ Measuring sound levels in decibels
• In the Richter scale, the independent variable is the amplitude of seismic waves, and the dependent variable is the magnitude.
• Logarithmic functions simplify scenarios with inverse exponential relationships.

  True
• Match the feature with its corresponding type of regression:

  Straight line relationship ↔️ Linear Regression

  Curvilinear relationship ↔️ Logarithmic Regression
• A positive correlation $r$ indicates that as $x$ increases, $y$ also increases
• A correlation coefficient between 0.8 and 1.0 is considered very strong
• Steps to evaluate the accuracy of a logarithmic regression model using the correlation coefficient $r$:


  1️⃣ Calculate the correlation coefficient $r$ for the model

  2️⃣ Determine the strength and direction of the correlation

  3️⃣ Interpret the correlation coefficient based on predefined ranges

  4️⃣ Assess the reliability of the model
• The coefficients $a$ and $b$ in the logarithmic function are determined by regression techniques.


  True
• A higher correlation coefficient in a logarithmic regression model indicates more reliable predictions.


  True
• Logarithmic functions are the inverse of exponential
• Logarithmic functions have a vertical asymptote at $x =$$0$
• The range of the exponential form $x =$$b^{y}$ is $y > 0$
• Logarithmic functions are useful for modeling situations where the rate of change decreases as the quantity increases.


  True
• What type of functions are logarithmic functions the inverse of?

  Exponential functions
• Where does a logarithmic function have a vertical asymptote?

  $x =$$0$
• What does the pH calculation model determine?

  Acidity of a solution
• In the pH calculation scenario, the independent variable is hydronium ion concentration
• In the Richter scale, the amplitude of seismic waves is the independent variable.


  True
• The domain of a logarithmic function is all real numbers.

  False
• The range of an exponential function is $y > 0$.


  True