Unit 3B Logarithms, Equations, and Regression Study Guide

Understanding Logarithmic Equivalence (Question 2): * The task requires identifying which logarithm among a provided list is equivalent to a specified target logarithm. * Task Requirements: * Check the box corresponding to the equivalent expression. * Write the numerical value of the logarithm in the designated "Equivalent Value" box.
Comparison of Specific Logarithmic Expressions (Question 3): * The evaluation focuses on determining equivalence or distinguishing between the following expressions: * $log_{16}(2)$ , which represents the power to which $16$ must be raised to obtain $2$ : $16^{1/4} = 2$ , so $log_{16}(2) = 0.25$ . * $log_{2}(16)$ , which represents the power to which $2$ must be raised to obtain $16$ : $2^4 = 16$ , so $log_{2}(16) = 4$ .

General Instructions for Solving Equations (Questions 4-9): * Precision Requirements: When calculations result in irrational or repeating decimals, answers must be rounded to the nearest ten-thousandths place (four decimal places). * Verification Step: Students are explicitly required to check for extraneous solutions. An extraneous solution is a root of a simplified form of an equation that does not satisfy the original equation (often occurring in logarithmic equations where the argument must be greater than zero).

Pizza Cooling Scenario (Question 10): * Context: A pizza is removed from an oven and undergoes a cooling process. * Initial Parameters: * Initial Temperature ( $T_0$ ): $415^\circ \text{F}$ . * Cooling Rate ( $r$ ): $4\%$ per time unit (expressed as $0.04$ in decimal form). * Target Temperature: $220^\circ \text{F}$ . * Mathematical Modeling: The problem requires the use of an exponential decay model to calculate the specific duration (number of minutes) required for the temperature to drop to the target level. * Formulaic Framework: The general model for exponential decay is typically represented as $y = a(1 - r)^t$ , where: * $y$ is the final amount ( $220^\circ \text{F}$ ). * $a$ is the initial amount ( $415^\circ \text{F}$ ). * $r$ is the decay rate ( $0.04$ ). * $t$ is the time in minutes.

Scenario Background: Math Geek Tech Stock Launch: * The firm Math Geek Tech is preparing to go public on the Stock Exchange. * Objective: Market analysts intend to create a regression model where the stock price is a function of time. * Variables: * Independant Variable ( $x$ ): Number of days the stock has been on the market. * Dependent Variable ( $y$ ): Stock price in dollars ( $USD$ ).
Regression Model Requirements (Question 11): * Type: A logarithmic regression equation must be derived based on provided data points. * Precision: All coefficients and constants within the regression model must be rounded to the nearest ten-thousandths.
Predictive Analysis (Questions 12-13): * Value Prediction: Determine the predicted stock price ( $y$ ) exactly $14$ days after the market launch ( $x = 14$ ). * Threshold Identification: Calculate the exact number of days ( $x$ ) it will take for the stock price to reach a value of $\$250$ .