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 logb x is equal to the exponent to which the base, b, is raised to produce product x ● Exponential equations can be written in logarithmic form, and vice versa ○ y = b^x <-> x = logby ○ y = logb X <-> 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 ● By altering the base, exponential functions and expressions can be stated in several ways. ● 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 than the earthquake intensity.