MHF4U Grade 12 Advanced Functions – Logarithms Test

Exponential Functions and its Inverse

  • an exponential function of the form y = bx , b > 0, b not equal 1, has
    • a repeating pattern of finite differences
    • a rate of change that is increasing proportional to the function for b > 1
    • a rate of change that is decreasing proportional to the function for 0 < b < 1
  • An exponential function of the form  y = bx , b > 0, b not equal 1,
    • has a domain X E R
    • a range Y E R, Y > 0
    • a y-intercept of 1
    • has a horizontal asymptote at y = 0
    • is increasing on its domain when b > 1
    • is decreasing on its domain when 0 < b < 1
  • The inverse of  y = bx is a function that can be written as x = by.
    • has a domain of X E R, x > 0
    • a range of Y E R
    • a x-intercept of 1
    • has vertical asymptote at x = 0
    • is a reflection of  y = bx about the line y = x
    • is increasing on its domain when b > 1
    • is decreasing on its domain when 0 < b < 1

 

 

Logarithms

  • a logarithmic function is the inverse of the exponential function
  • The value of logbx is equal to the exponent to which the base, b, is raised to produce product x
  • Exponnetial equations can be written in logarithmic form, and vice versa
    • y = b^x   <->  x = logby
    • y = logbX   <->   x = b^y
  • Exponential and logarithmic functions are defined only for positive values of the base that are not equal to one. In other words, b not = 1, and x > 0.
  • The logarithm of x to base 1 is only valid when x = 1, in which base y has an infinite number of solutions and is not a function.
  • Common logarithms are logarithms wit ha base of 10. It is not necessary to write the base for common logarithms: logx means log base 10 x.

 

 

Transformations of logarithmic Functions

  • The techniques for applying transformations to logarithmic functions are the same for those used for other functions:
    • y = log x + c
      • translate up c units if c > 0
      • translate down c units if c < 0
    • y = log( x – d)
      • translate right d units if d > 0
      • translate left d units if d < 0
    • y = a log x
      • stretch vertically by a factor of |a| if |a| > 1
      • Compress vertically by a factor of |a| if |a| < 1
      • Reflect in the x-axis if a < 0
    • y = log (kx)
      • compress horizontally by a factor of |1/k| if |k| < 1, k not = 0.
      • Reflect in the y axis if k < 0.
  • When all transformations are combined, they follow the form:

 

f(x) = a log[k(x-d)] + c

 

Power of logarithms

  • The power of logarithms states that logbxn = n logbx for b > 0, b not = 1, x > 0, and n ER
  • Any logarithm can be expressed in terms of common logarithms using the change of base formula:
  • log base b m = log m / log b, b > 0, b not = 1, m > 0

 

Exponential Functions

  • Exponential functions and expressions can be expressed in different ways by changing the base
  • Changing the base of one or more exponential expressions is a useful technique for solving exponential equations

 

 

Solving Exponential Functions

  • An equation maintains balance when the common logarithm is applied to both sides
  • The power of logarithms is a useful tool for solving a variable that appears as a part of an exponent
  • When a quadratic equation is obtained, methods such as factoring and applying the quadratic formula may be useful.
  • Some algebraic methods of solving exponential functions lead to extraneous roots, which are not valid solutions to the original equation

 

 

Laws of logarithms

  • The product law of logarithms states that logbX + logb Y = logb(XY) for b > 0, b not = 1, x > 0, y > 0
  • The quotient law of logarithms states that logbX – logbY = logb(X/Y) for b > 0, b not = 1, x > 0, y > 0

 

 

Applications of Exponential/ Logarithmic Functions

  • Interest rates:

A = (i + 1)t

Where A is the amount, i is the interest, and t is the time.

 

  • Population

P = A1 (i)t

Where P is the population, A1is the initial amount, i is the amount increase, and t is time.

 

  • Half Life:

Ao = Ai (1/2)t/h

Where Ao is the final amount, Ai is the initial amount, t is time, and h is the time interval

 

  • pH Levels:

pH = -log (H+)

Where H+ is the concentration of hydrogen ions

Every Integer increment in pH is 10 times more acidic

 

  • Sound Intensity:

L = 10log(I / Io)

Where L is the loudness, I is the intensity, and Io is the sound that is barely audible

Every Integer increment in L, decibels, is 10 times more intense

 

  • Earthquakes:

M = log(E/ Io)

Where M is the Richter Number, E is the earthquake’s intensity, and Io is the intensity of a referenced earthquake.

Every Integer increment in M, Richter readings, is 10 times greater the earthquake intensity.