Logs and Exponentials

• What is the logarithmic form of an exponential function $y =$$a^x?$


  $x =$$\log_a y$
• How is $y =$$a^x$ expressed in logarithmic form?


  $x =$$\log_a y$
• If $a^3 =$$8,$ what is the value of $a?$


  $a =$$2$
• To determine the equation of a logarithmic function from its graph, what information should be substituted into the equation?

  Information from the graph, such as coordinates and characteristics.
• What two coordinates are needed to sketch the graph of an exponential function from its equation?

  • Coordinate when $x =$$0$
  • Coordinate when $x =$$1$
• What two coordinates are needed to sketch the graph of a logarithmic function from its equation?

  • Coordinate when $y =$$0$
  • Coordinate when $y =$$1$
• How is the graph of an inverse function related to the original function?

  It is reflected in the line $y =$$x.$
• What is the first step in sketching the graph of an inverse function?

  Sketch the graph of the function.
• How is the inverse graph drawn as a reflection of the original function?

  • Reflect each coordinate in the line $y =$$x.$
  • Annotate each reflected coordinate on the inverse graph.
• How are the graph transformations of logarithmic and exponential functions related to the transformations of other functions?

  They follow the same principles as the functions seen in section 5.
• State the First Law of logarithms.

  $\log_a b +$$\log_a c =$$\log_a bc$
• What is the Second Law of logarithms?

  $\log_a b - \log_a c =$$\log_a \frac{b}{c}$
• What is the Third Law of logarithms?

  $\log_a b^r =$$r \log_a b$
• What is the Fourth Law of logarithms?

  $\log_a 1 =$$0$
• What is the Fifth Law of logarithms?

  $\log_a a =$$1$
• How can the expression $\log_5 (x + 1) +$$\log_5 (x - 3)$ be simplified using the First Law of logarithms?


  $\log_5 (x + 1)(x - 3)$
• State the Second Law of logarithms.

  $\log_a b - \log_a c =$$\log_a \frac{b}{c}$
• What is the Third Law of logarithms?

  $\log_a b^n =$$n \log_a b$
• What functions are primarily used when working with exponential growth and decay?

  • Exponential function $e^x$ or $\exp x$
  • Natural log function $\ln x$ or $\log_e x$
• How is the initial value determined when working with exponential growth and decay?

  Substitute given values into the equation to determine the initial value.
• How is half-life calculated in exponential decay?

  Make the equation equal to one half.
• If $500 =$$A_o e^{-0.004 \times 100},$ how can $A_o$ be calculated?


  $A_o \approx 27300$
• How is the half-life equation related to the decay function?

  When the quantity decays to half its initial value.
• What is the half-life of the substance in the function $A_t =$$A_o e^{-0.004t}?$


  173 years.
• What are the two types of exponential functions considered in experimental data questions?

  $y =$$kx^n$ and $y =$$ab^x.$
• How is $y =$$kx^n$ expressed in logarithmic form when log₄y is plotted against log₄x?


  $\log_4 y =$$n \log_4 x +$$\log_4 k$
• What are the two methods to find the constants $k$ and $n$ in the function $y =$$kx^n?$


  • Calculate $m$ from the slope.
  • Substitute coordinates and solve simultaneously.
• What is the slope $m$ obtained in the logarithmic form of $y =$$kx^n?$


  2.
• What is the value of $k$ obtained by substituting $m =$$2$ into equation (A) for $y =$$kx^n?$


  64.
• How is $y =$$ka^x$ expressed in logarithmic form when logy is plotted against x?


  $\log y =$$x \log a +$$\log k$
• What is the value of $k$ obtained from $\log k =$$2?$


  9.
• How can the gradient $m$ be determined for the logarithmic form of $y =$$ka^x?$


  By using the gradient formula or substitution of $log k$ and one coordinate.
• What is the equation derived using the gradient formula for $y =$$ka^x?$


  $11 =$$3 \log a +$$2$
• What equation is obtained by simplifying $11 =$$3 \log a +$$2?$


  $9 =$$3 \log a$
• What is the value of $k$ obtained from $\log_3 k =$$2?$


  9.
• What is the value of $a$ obtained from $3 =$$\log_3 a?$


  27.
• What are the two methods to determine $k$ and $a$ in the function $y =$$ka^x?$


  • Determine $k$ from the y-intercept.
  • Use the gradient formula with coordinates to find $a.$
• How is $y =$$ka^x$ expressed in logarithmic form when $\log_3 y$ is plotted against $x?$


  $\log_3 y =$$x \log_3 a +$$\log_3 k$