UNIT A

PreCalculus Review

Unit Objectives 1. Review characteristics of fundamental functions

(R)

2. Review/Extend application of function models (R/E)

3. Introduce new function concepts pertinent to Calculus (N)

A8 - Logarithmic Functions

Calculus - Santowski

Lesson Objectives 1. Simplify and solve logarithmic expressions 2. Sketch and graph logarithmic fcns to find graphic

features 3. Explore logarithmic functions in the context of

calculus related ideas (limits, continuity, in/decreases and its concavity)

4. Logarithmic models in biology (populations), business (profit, cost, revenue)

Fast Five 1. Solve ln(x + 2) = 3 2. Sketch f(x) = -ln(x + 2) 3. State the domain of f(x) = (ln(x - 1))0.5

4. Evaluate log232 + log31/81+ log0.2516 5. Sketch the inverse of f(x) = 3 - log2x 6. Find the domain of log3(9 - x2) 7. Evaluate log3324 - log34

8. Expand using LoL

9. Evaluate log79 + log35 using GDC 10. Solve 3x = 11

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log2

a4

bc

Explore Using your calculator for confirmation, and

remembering that logarithms are exponents, explain why it is predictable that:

(a) log 64 is three times log 4; (b) log 12 is the sum of log 3 and log 4; (c) log 0.02 and log 50 differ only in sign.

(A) Logarithmic Fcns & Algebra (1) Find the inverse of f(t) = 67.38(1.026)t

(2) Solve e3-2x = 4 (3) Solve log6x + log6(x - 5) = 2

(3) Express as a single log

(4) Solve log2x + log4x + log8x = 11

(5) Simplify

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1

2ln x − 4 ln y + ln x 2 +1( )

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lnx

x +1

⎛

⎝ ⎜

⎞

⎠ ⎟

(B) Logarithmic Fcns &Graphs Be able to identify asymptotes, intercepts, end

behaviour, domain, range for y = logax

Ex. Given the function y = log2(x - 1) - 2, determine the following: - domain and range - asymptotes - intercepts - end behaviour - sketch and then state intervals of increase/decrease as well

as concavities

(B) Logarithmic Fcns & Graphs

Ex 3. Given the functions f(x) = logx and g(x) = x1/3,

(a) when is f(x) > g(x) (b) when is 100f(x) < g(x) (c) Which function increases faster,

100f(x) or g(x)?

(C) Logarithmic Fcns & Calculus Concepts Now we will apply the concepts of limits,

continuities, rates of change, intervals of increase/decreasing & concavity to exponential function

Ex 1. Graph

From the graph, determine: domain, range, max and/or min, where f(x) is increasing, decreasing, concave up/down, asymptotes€

f (x) = ln x 2 −1( )

(C) Logarithmic Fcns & Calculus Concepts Ex 2. Evaluate the following limits numerically or

algebraically. Interpret the meaning of the limit value.

Then verify your limits and interpretations graphically.

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limx→∞

ln x + 4( )

limx→−4 +

ln x + 4( )

limx→π −

ln sin x( )

limx→1+

log10 x2 − x( )

(C) Logarithmic Fcns & Calculus Concepts Ex 3. Given the function :

(i) find the intervals of increase/decrease of f(x) (ii) is the rate of change at x = 2 equal to/more/less than

the rate of change equal to/greater/less than the rate at x = 1?

(iii) find intervals of x in which the rate of change of the function is increasing. Explain why you are sure of your answer.

(iv) where is the rate of change of f(x) equal to 0? Explain how you know that?

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f (x) =ln x

x

(C) Logarithmic Fcns & Calculus Concepts Ex 4. Given the function , find the average rate of change of

f(x) between:

(a) 1 and 1.5 (b) 1.4 and 1.5 (c) 1.499 and 1.5 (d) predict the rate of change of the fcn at x = 1.5 (e) evaluate limx1.5 f(x). (f) Explain what is happening in the function at x = 1.5 (g) evaluate f(1.5) (h) is the function continuous at x = 1.5? (i) is the function continuous at x = 0?

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f (x) =ln x

x

(D) Applications of Logarithmic Functions

The population of Kenya was 19.5 million in 1984 and was 32.0 million in 2004. Assuming the population increases exponentially, find a formula for the population of Kenya as a function of time. Then using logs, find the doubling time of Kenya’s population

(E) Internet Links Logarithm Rules Lesson from Purple Math

College Algebra Tutorial on Logarithmic Properties from West Texas AM

You can try some on-line word problems from U of Sask EMR problems and worked solutions

More work sheets from EdHelper's Applications of Logarithms: Worksheets and Word Problems

(F) Homework

Text pages 113-117 (1) Evaluate logs, Q19,20,23 (1) properties of logs, Q31,33,35 (2) solving eqns, Q39,41,51,53,60 (3) apps, Q71,77,81