y = x

1 2 x

1

2

–1

–2

y

MHF 4U Lesson 7.1 THE LOGARITHMIC FUNCTION

The inverse of the Exponential Function y = bx, is obtained by interchanging the x and

the y coordinates. The inverse of y = bx is x = b

y. The graph of the inverse is a reflection

in the line y = x.

Since the exponential function is only

defined for b > 0, it makes sense that the

inverse function is only defined for b > 0. We

can see from the graph that the domain of x

= by is x > 0. Since this inverse function is so

important in mathematics, it is given its own

name.

It is called the Logarithmic Function

And it is written y = log

bx and read as

“y equals the log of x to the base b.”

the log function is defined only for

x > 0 and b > 0.

Properties of the Logarithmic Function y = logbx.

The base b is positive.

The x–intercept is 1.

The y–axis is a vertical asymptote.

The domain is: D = {x| x > 0, x R}.

The range is: R = {y|y R}.

The function is increasing if b > 1.

The function is decreasing if 0 < b < 1.

While any number can be used as the base, the most common base is 10. As a result, any

logarithm with a base of 10 is called a common logarithm. Since it is so common, log10 x

is usually written as log x. Calculators are programmed in base 10, today, you cannot

use your calculator for any other base. (You will learn later about another base for which

calculators are programmed.)

Exponential Form Logarithmic Form

x = by logb x = y b > 0 and b ≠ 1

The logarithm of a number x with a base b is the exponent to which b must be raised to

yield x.

y = logbx

y = bx

As with the exponential function, there are two possible variations of the graph of the

logarithmic function.

1 2 3 x

1

2

–1

–2

y

1 2 3 x

1

2

–1

–2

y

y = logbx, b > 0

y = logbx, 0 < b < 1

Ex. 1 Rewrite in exponential form.

a) log4 64 = 3 b) loga b = c c) y = log x

Ex. 2 Rewrite in logarithmic form.

a) 52 = 25 b) ab = c c) 3-2 = 9

1

Ex. 3 Evaluate each of the following.

a) log3 81 b) log2 8

1 d) log 0.01

Pg. 451 # 1 – 6

MHF 4U INV 7.2 Transformations of the Logarithmic Function

How do transformations affect the graph of a logarithmic function f(x) = alog (kx – d) + c ?

For this investigation, you will use the common logarithm f(x) = log x and a TI-83 calculator.

A: The Effects of c and d in f(x) = log (x – d) + c.

1. Graph the function xxf log)( on your graphing calculator

2. Predict what the graph of 3)log()( xxf will look like.

3. Verify using the TI-83. Sketch the graph of 3log)( xxf below. The graph of )log()( xxf is given.

4. Predict what the graph of )2log()( xxf will look like.

5. Verify using the TI-83. Sketch the graph of )2log()( xxf below. The graph of )log()( xxf is given.

1 2 3 4 5 6 7 8 9–1–2–3–4–5–6–7–8–9 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

1 2 3 4 5 6 7 8 9–1–2–3–4–5–6–7–8–9 x

1

2

3

4

5

6

7

–1

–2

–3

–4

–5

–6

–7

y

To draw logs with a base other than 10 on

the TI-83, you must type in

log(argument)/log(base)

Ex. To graph log2x, type log(x)/log(2)

f(x) = log x

B: The Effects of a and k in f(x) =a log (kx).

1. Graph the function xxf log)( on your graphing calculator

2. Predict what the graph of xxf log2)( will look like.

3. Verify using the TI-83. Sketch the graph of xxf log2)( below. The graph of )log()( xxf is given.

4. Predict what the graph of )3log()( xxf will look like.

5. Verify using the TI-83. Sketch the graph of )3log()( xxf below. The graph of )log()( xxf is given.

SO, what does:

a do?

k do?

d do?

c do?

1 2 3 4 5 6 7 8 9–1–2–3–4–5–6–7–8–9 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

1 2 3 4 5 6 7 8 9–1–2–3–4–5–6–7–8–9 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

Ex. 1 a) Sketch the graph of the function 5)2log()( xxf . The graph of )log()( xxf is given.

b) state the key features of the function.

(i) the domain

(ii) the range

(iii) x-intercept, if it exists.

(iv) y-intercept, if it exists.

(v) equation of any asymptote.

Ex. 2 Sketch the graph of each function and identify the key features of each graph.

a) )3log(5)( xxf The graph of )log()( xxf is given.

(i) the domain

(ii) the range

(iii) x-intercept, if it exists.

(iv) y-intercept, if it exists.

(v) equation of any asymptote.

1 2 3 4 5 6 7 8 9–1–2–3–4–5–6–7–8–9 x

1

2

3

4

5

6

7

–1

–2

–3

–4

–5

–6

–7

y

1 2 3 4 5 6 7 8 9–1–2–3–4–5–6–7–8–9 x

1

2

3

4

5

6

7

–1

–2

–3

–4

–5

–6

–7

y

b) 4)2log()( xxf The graph of )log()( xxf is given.

(i) the domain

(ii) the range

(iii) x-intercept, if it exists.

(iv) y-intercept, if it exists.

(v) equation of any asymptote.

Ex. 3 Sketch the graph of the function )42log()( xxf . The graph of )log()( xxf is given.

1 2 3 4 5 6 7 8 9–1–2–3–4–5–6–7–8–9 x

1

2

3

4

5

6

7

–1

–2

–3

–4

–5

–6

–7

y

1 2 3 4 5 6 7 8 9–1–2–3–4–5–6–7–8–9 x

1

2

3

4

5

6

7

–1

–2

–3

–4

–5

–6

–7

y

Pg. 457 # 1 – 4, 7, 9

MHF 4U Lesson 7.3 Evaluating Logarithms

In the last lesson, when we evaluated logarithms other than the common logarithm we created an equation to

solve. There is a quicker more efficient method of evaluating logs.

For example, to evaluate log2 64, we let y = log2 64

wrote the equation in exponential form 2y = 64

wrote 64 with a base of 2 2y = 26

solved for y y = 6

and had evaluated the logarithm.

Notice that all we did was write the argument with the same base as the logarithm in order to evaluate.

So, we can skip steps and and immediately write the argument with the same base as the logarithm.

ie: log2 64

= log2 26 and the answer is the exponent in the argument.

= 6

Ex. 1 Evaluate each of the following.

a) log3 2187 b) log5 125 c)

64

27

4

3log

d) log 100 000 e) log9 729 f) log8 4096

1

g) log2 3 512 h) log6 1296 i) log7 (-343)

What if the argument cannot be easily written with the same base as the logarithm?

For example, what if you had to evaluate log5 314?

There are a number of ways we can solve this problem.

Use a graphing calculator. ------ not everyone has access to a graphing calculator at all times.

Estimate the exponent to which 5 must be raised in order to get 314 by guess and check.

- can be time consuming

Change the base to 10 so that we can use a scientific calculator to find the estimate.

If y = logx a

we can rewrite it in exponential form.

take the log10 of both sides

use laws of logs to rearrange

Solve for y

In general, logx a = x

a

b

b

log

log, where b is any base you wish. ( 10 is most commonly used.)

Ex. 2 Evaluate log5 314 correct to 3 decimal places.

Ex. 3 Evaluate log7 2.391 correct to 3 decimal places.

Pg. 466 # 1 - 10

MHF 4U Lesson 7.4 LAWS OF LOGARITHMS

Basic Properties of Logarithms PROOFS

1. logb 1 0 1.

logb 1 0

2. logb b 1

3. logb bx x

4. blog

bx x

2. logb b 1 3.

logb bx x

4. blog

bx x

Laws of Logarithms When x > 0, y > 0, and r is a real number,

1. loga xy loga x loga y

2.

logax

y

loga x loga y

3. loga xr rloga x

PROOFS

1. If m = loga x , then am = x and if n = loga y,

then an = y

2. If m = loga x , then am = x and if n = loga y, then a

n = y

3. If m = logax, then x = am

If we raise each side to the power r

Examples

1. Find the value of each of the following.

a) log354 + log3

2

3 b) log2144 – log29 c) log42 + log432 d) 5

1

3 log5 8

2. Simplify log3

81

27. 3. Write 25log yxb in terms of xblog and yblog .

4. Describe the transformation(s) that must be applied to xy 2log to obtain:

a) y = 32log xy b)

8log 2

xy

5. Write as a single logarithm.

a) 2log232 + log26 – 3log23 b) 3log(x + 3) – 2log(x – 1)

Pg. 475 # (1 – 7)doso, 8, (9 – 11)doso, 12, 13

LAWS of LOGS SUMMARY

MHF 4U Lesson 7.5 Solving Exponential Equations

When solving an exponential equation, we want to manipulate the equation into the form ax = ay.

If the bases are the same, then the exponents must be equal. x = y.

So to solve an exponential equation, write the powers with the same base and work with the

exponents.

If the bases cannot easily be written with the same base, use other methods.

Ex. 1 Solve each of the following.

a) 4x + 1 = 2x – 1 b) 93x + 1 = 27x

c) 21633 2 xx d) 145833 23 xx

e) x

x

x

24

21

1

f) 462 932

xx

Ex. 2 Solve correct to 3 decimal places.

a) 8x = 51 b) 3(2.1x) = 65

c) 52x – 3 = 21(51x) d) 132x – 5 = 7(43x + 1)

Ex. 3 300 mg of iodine is stored for 30 d. At the end of this time, 12 mg remain. Find the half-life correct to

2 decimal places.

Pg. 485 # 1 – 3, 5, 8ace, 9, 10

MHF 4U Lesson 7.6 Solving Logarithmic Equations

We can solve logarithmic equations in a similar way to the method by which we solved exponential equations.

With exponential equations, if we manipulated the equation into the form ax = ay, we knew that x = y.

ie: if the bases were the same, the exponents must be equal.

We can use the same principle to solve logarithmic equations. As long as the bases are the same, the arguments

must also be equal.

If loga x = loga y, then x = y.

Ex.1 Solve each of the following and reject any extraneous roots.

a) log (x + 4) = 1 b) log5 (2x – 3) = 2

c) log (x + 5) =2log (x - 1)

d) log (x – 1) – 1 = -log (x + 2)

e) log 3 2 48xx = 3

2

f) logx 5 = 2

Pg. 491 # (1 – 8)doso, 9, 10, 12

MHF 4U Lesson 7.7 Applications of Exponential Functions

Various Forms of Exponential Functions

Compound Interest:

niPA )1( , where A is the accumulated amount

P is the original principal invested i is the interest rate per compounding period (annual rate / #of compound periods per year) n is the number of compounding periods

Doubling period is the period of time required for a quantity to grow to twice its original amount.

Exponential Growth (involving a doubling period):

dt

oAA )2( , where A is the total amount or number

Ao is the initial amount or number 2 is the growth factor t is the time, d is the doubling period

Half-life is the time required for a material to decay to one-half of its original mass or quantity.

Exponential Decay (involving half-life):

h

t

oAA

2

1, where A is the remaining mass of the decayed material

Ao is the original mass of the material

2

1is the decay factor

t is the time h is the half-life

Exponential Function:

xbay )( , where y is the total amount or number

a is the initial amount or number b is the growth factor or decay rate x is the number of growth periods or decay period For an increase problem, )1( ib where i is the rate of increase

For a decrease problems (depreciation), )1( ib where i is the rate of decrease

Ex. 1 The population of the town of Euler’s population over the years 1960 – 2005 is shown below.

Use the information in the table to answer the following questions.

a) Create an exponential model for this data, where n represents the number of years since 1960 and

P represents the population of Euler.

b) Use the equation to determine the population of Euler in 2025.

c) Use the equation to determine when the population:

(i) was 200. (ii) will be 110 000.

Ex. 2 $5000 is invested in an account that pays 5.2%/a compounded semi-annually.

a) Find a model for the amount of money in the account after x years.

b) How many years will it take the investment to reach $14 000?

Ex. 3 Yeast cells increase their numbers exponentially by a process called budding. They duplicate themselves

every half hour. If the initial number of yeast cells is 500:

a) How many cells would there be after 6 hours?

b) How long would it take yeast cell population to reach 50 million cells?

The formula used by Charles Richter to define the magnitude of an earthquake is

oI

IM log ,

Where I is the intensity of the earthquake being measured, Io is the intensity of a reference

Earthquake and M is the number used to measure their intensity.

The formula to compare sounds is

oI

IL log10 , where I is the intensity of the sound being measured,

Io is the intensity of a sound at the threshold of hearing, and L is the loudness measured in decibels.

The formula for defining the acidity of a liquid on a pH scale is ]log[ HpH , where H+

is the

concentration of the hydrogen ion in moles per litre,

Ex. 4 An earthquake of magnitude 7.5 on the Richter scale struck Guatemala on February 4, 1976.

On October 2, 1993 and earthquake of magnitude 6.4 struck Maharashra, India. Compare the intensities

of the two earthquakes.

Ex. 5 How many times more intense is the sound of normal conversation (60 dB) than the sound

of a whisper (30 db)?

Ex. 6 The pH of a fruit juice is 3.10. What is the hydrogen ion concentration of the fruit juice?

WS 7.7 doso

Pg. 499 # 1 – 4, (5, 6)ac, 7, 8