Logarithmic FunctionsCHAPTER 7

Chapter 7

7.1 – CHARACTERISTICS OF

LOGARITHMIC FUNCTIONS WITH

BASE 10 AND BASE E

LOGARITHMS

Logarithms are a new operation that we will learn. It is like multiplication, exponents or trigonometric functions. They give us a way to solve equations that have variables in the exponent.

Ex. 120 = 10y

A logarithmic function is a function of the form:

y = alogbx

where b > 0, b ≠ 1, and a ≠ 0, and a and b are real numbers.

LOGARITHMIC FUNCTIONS

f(x) = log10x f(x) = 2log10x f(x) = 5log10x

LOGARITHMIC FUNCTIONS

The expression log10x is known as the common logarithm or a logarithm with a base of 10. The expression is often written without the 10, so the two functions y = log10x and y = logx are equivalent.

y = log10x Is equivalent to

x = 10y

A logarithm with base e is called the natural logarithm and is written as lnx. The functions y = logex, y = lnx, and x = ey are equivalent.

EXAMPLE

Predict the x-intercept, the number of y-intercepts, the end behaviour, the domain, and the range of the following function:

y = 15logxUse the equation of the function to make your predictions. Verify your predictions using your graphing calculator.

EXAMPLE

Predict the x-intercept, the number of y-intercepts, the end behaviour, the domain, and the range of the following function:

y = –4lnxUse the equation of the function to make your predictions. Verify your predictions using your graphing calculator.

EXAMPLE

Which function matches each graph below? i) y = 5(2)x ii) y = 2(0.1)x iii) y = 6logx iv) y = –2lnx

Independent practice

PG. 420-425, #1, 2, 5, 7, 8, 12, 15

Chapter 7

7.2 – EVALUATING LOGARITHMIC EXPRESSIONS

EXAMPLE

Determine the value of y in each exponential equation. Verify your answer.a) 81 = 10y b) 25 = ey

EXAMPLE

Evaluate each logarithmic expression. a) log216 b) log464

EXAMPLE

Evaluate each logarithmic expression without technology.a) b) c)

EXAMPLE

Which expression has the greater value?

A: log216 + log22 B: log21 – log2(1/8)

EXAMPLE

The pH scale in chemistry is used to measure the acidity of a solution. The pH scale is logarithmic, with a base of 10. A logarithmic scale is useful for comparing numbers that vary greatly in size. The pH, p(x), is defined by the equation

p(x) = –logxwhere the concentration of hydrogen ions, x, in a solution is measured in moles per litre (mol/L).

a) The hydrogen ion concentration, x, of a solution is 0.0001 mol/L. Calculate the pH of the solution.

EXAMPLE

The pH scale in chemistry is used to measure the acidity of a solution. The pH scale is logarithmic, with a base of 10. A logarithmic scale is useful for comparing numbers that vary greatly in size. The pH, p(x), is defined by the equation

p(x) = –logxwhere the concentration of hydrogen ions, x, in a solution is measured in moles per litre (mol/L).

b) Lemon juice has a pH of 2. What is the hydrogen ion concentration of lemon juice?

EXAMPLE

The pH scale in chemistry is used to measure the acidity of a solution. The pH scale is logarithmic, with a base of 10. A logarithmic scale is useful for comparing numbers that vary greatly in size. The pH, p(x), is defined by the equation

p(x) = –logxwhere the concentration of hydrogen ions, x, in a solution is measured in moles per litre (mol/L). c) In terms of hydrogen ion concentration, how much more acidic is Solution A, with a pH of 1.6, than Solution B, with a pH of 2.5? Round your answer to the nearest tenth.

Independent practice

PG. 436-438, #1-12, 14, 15, 17

To hand in:

Chapter 77.3 – LAWS OF

LOGARITHMS

LOGARITHM IDENTITIES

logbm + logbn = logb(mn)

EXAMPLE

Use exponent laws to explain why log12 + log2 is equivalent to log24

Remember that:

x = log12 10x = 12

y = log2 10y = 2

10x ∙ 10y = 12 ∙ 2 = 24

Power laws say that:10x+y = 24x + y = log24

Substitute:log12 + log2 = log24

EXAMPLE

Simplify and then evaluate each logarithmic expression.a) log25 + log26.4 b) log5100 – log54 c) log3275

Try these:a) log248 – log23 b) log467

EXAMPLE

Write each expression as a single logarithm, and then evaluate.a) log318 + log3(1.5) b) log540 – 3log52

Independent practice

PG. 446-448, #1-7, 11, 12, 15, 16

Chapter 7

7.4 – SOLVING EXPONENTIAL

EQUATIONS USING LOGARITHMS

EXAMPLE

Yvonne has a balance of $3215 in her savings account. This account pays 2.4% interest per year, compounded annually. The compound interest formula is:

A = P(1 + i)n

Where A represents the future value, P represent the principal, I represents the interest rate applied each compounding period, and n represents the number of compounding periods. How long will it take for Yvonne’s balance to reach $5000?P = 3215

i = 0.024A = 5000

A = P(1 + i)n

5000 = 3215(1 + 0.024)n

Take the log of both sides:

Use the log laws

It will take 19 years.

EXAMPLE

Solve the following exponential equation, to three decimal places:3x–1 = 20

EXAMPLE

Solve 2x–1 = 3x+1, and round your answer to three decimal places.

EXAMPLE

Evaluate log2100 to three decimal places.

Independent practice

PG. 455-458, #4, 5, 6, 7, 9, 10, 12, 15,

16

Chapter 7

7.5 – MODELLING DATA USING

LOGARITHMIC FUNCTIONS

EXAMPLE

The flash on most digital cameras requires a charged capacitor in order to operate. The percent charge, Q, remaining on a capacitor was recorded at different times, t, after the flash had gone off. The t.5 flash duration represents the time until a capacitor has only 50% of its initial charge. The t.5 flash duration also represents the length of time that the flash is effective, to ensure that the object being photographed is properly lit.

a) Construct a scatter plot for the given data.

EXAMPLE

b) Determine a logarithmic model for the data.

The flash on most digital cameras requires a charged capacitor in order to operate. The percent charge, Q, remaining on a capacitor was recorded at different times, t, after the flash had gone off. The t.5 flash duration represent the time until a capacitor has only 50% of its initial charge. The t.5 flash duration also represents the length of time that the flash is effective, to ensure that the object being photographed is properly lit.

EXAMPLE

The flash on most digital cameras requires a charged capacitor in order to operate. The percent charge, Q, remaining on a capacitor was recorded at different times, t, after the flash had gone off. The t.5 flash duration represent the time until a capacitor has only 50% of its initial charge. The t.5 flash duration also represents the length of time that the flash is effective, to ensure that the object being photographed is properly lit.c) Use your logarithmic model to determine the t.5 flash duration to the nearest hundredth of a second.

EXAMPLE

Caffeine is found in coffee, tea, and soft drinks. Many people find that caffeine makes it difficult for them to sleep. The following data was collected in a study to determine how quickly the human body metabolizes caffeine. Each person started with 200 mg of caffeine in her/his bloodstream, and the caffeine level was measured at various times.Enter the data on the handout into your calculator.

a) Determine the equation of the logarithm regression function for the data representing time as a function of caffeine level.

b) Determine the time it takes for an average person to metabolize 50% of the caffeine in his/her bloodstream. Round to the nearest tenth of an hour.

c) Paula drank a cup of coffee that contained 200 mg of caffeine at 10:00 am. How much caffeine will be in her bloodstream at 9:00 pm? Round your answer to the nearest milligram.

Independent practice

PG. 466-471, #2, 3, 6, 7, 9, 11.