Cards In This Deck

• $$\begin{gathered} \text{True or False?} \\ \text{The graph of }y=a^x\text{ (where }a>0,a\ne 1\text{) touches the x-axis at large negative values of x.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{The x-axis (}y=0\text{) is an asymptote. The curve approaches it but never touches or crosses it.} \end{gathered}$$
• $\text{When sketching the graph of }y=a^x\text{ (}a>0\text{), how do you determine if the function is increasing or decreasing?}$
  $$\begin{gathered} \text{- If }a>1\text{: The function is increasing (growth).} \\ \text{- If }0<a<1\text{: The function is decreasing (decay).} \end{gathered}$$
• $\text{How do you sketch the graph of }y=3(2^x)\text{ based on }y=2^x\text{?}$
  $$\begin{gathered} \text{Apply a vertical stretch with scale factor 3.} \\ \text{The y-intercept changes from }(0,1)\text{ to }(0,3)\text{. The asymptote remains }y=0\text{.} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{If }{a}^x=b\text{, then }x={\log }_{b}a\text{.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct definition: }x={\log }_{a}b \\ \text{(The base of the power stays the base of the log).} \end{gathered}$$
• $\text{What is the condition for the solution of }{a}^x=b\text{ to exist (where }a>0\text{)?}$
  $b\text{ must be positive (}b>0\text{).}$
• $$\begin{gathered} \text{True or False?} \\ ({\log }_{a}x{)}^k=k{\log }_{a}x \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ The Power Law only applies when the power is inside the logarithm argument:} \\ {\log }_a(x^k)=k{\log }_{a}x \\ ({\log }_{a}x{)}^k\text{ means "the value of log x multiplied by itself k times".} \end{gathered}$$
• $\text{When solving an equation like }2{\log }_{2}x+{\log }_2(x-1)=3\text{, what must be done to the number ’3’ before combining the logs?}$
  $$\begin{gathered} \text{Rewrite ’3’ as a logarithm with the same base.} \\ 3=3{\log }_{2}2={\log }_2(2^3)={\log }_{2}8 \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ {\log }_a(x+y)={\log }_{a}x+{\log }_{a}y \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{There is no law for the logarithm of a sum.} \\ \text{✅ The Multiplication Law is: }{\log }_a(xy)={\log }_{a}x+{\log }_{a}y \end{gathered}$$
• $\text{What are the restrictions on the base }a\text{ and the argument }x\text{ for the function }y={\log }_{a}x\text{ to be defined?}$
  $$\begin{gathered} \text{1. Base }a\text{: }a>0\text{ and }a\ne 1 \\ \text{2. Argument }x\text{: }x>0 \end{gathered}$$
• $\text{What is the value of }{\log }_{a}1\text{?}$
  ${\log }_{a}1=0$
• $$\begin{gathered} \text{True or False?} \\ \text{When solving for }n\text{ in the inequality }0.8^n<0.05\text{, the inequality sign flips when you take logarithms.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False (at the log step)} \\ \text{Taking logs (e.g., }\ln (0.8^n)<\ln (0.05)\text{) does not flip the sign. } \\ \text{However, the sign WILL flip when you divide by }\ln (0.8)\text{ because }\ln (0.8)\text{ is a negative number.} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \frac{{\log }_{a}x}{{\log }_{a}y}={\log }_a(x-y) \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ The Division Law applies to the log of a quotient, not the quotient of logs:} \\ {\log }_a\left(\frac{x}{y}\right)={\log }_{a}x-{\log }_{a}y \\ \text{Note: }\frac{{\log }_{a}x}{{\log }_{a}y}\text{ is actually }{\log }_{y}x\text{ (Change of Base Law).} \end{gathered}$$
• $\text{Which rule should you use to solve an equation involving logs with different bases, such as }{\log }_{2}x+{\log }_{4}x=3\text{?}$
  $$\begin{gathered} \text{Use the Change of Base Formula to unify the bases.} \\ {\log }_{a}x=\frac{{\log }_{b}x}{{\log }_{b}a} \end{gathered}$$
• $\text{What is the standard first step to solve an equation in the form }{p}^{2x}+p^x-12=0\text{?}$
  $$\begin{gathered} \text{Use a quadratic substitution.} \\ \text{Let }u=p^x\text{, so the equation becomes }{u}^2+u-12=0\text{.} \end{gathered}$$
• $\text{If given }y=a{x}^n\text{, how can you plot a straight line graph to find }a\text{ and }n\text{?}$
  $$\begin{gathered} \text{Take logs of both sides (usually base 10):} \\ \log y=\log (a{x}^n) \\ \log y=n\log x+\log a \\ \text{Plot }\log y\text{ vs }\log x\text{. Gradient = }n\text{, Intercept = }\log a\text{.} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \frac{1}{{\log }_{a}b}={\log }_a\left(\frac{1}{b}\right) \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Reciprocal Rule: }\frac{1}{{\log }_{a}b}={\log }_{b}a \\ \text{Note: }{\log }_a(\frac{1}{b})={\log }_a(b^{-1})=-{\log }_{a}b\text{. These are different results.} \end{gathered}$$