Math 30-1
Exponents & Logarithms
Specific Outcome: Students will demonstrate an understanding of logarithms.
Achievement Indicators:
Explain the relationship between logarithms and exponents.
Express a logarithmic expression as an exponential expression and vice versa. Determine, without technology, the exact value of a logarithm, such as . Estimate the value of a logarithm, using benchmarks, and explain the reasoning.
Specific Outcome: Students will demonstrate an understanding of the product, quotient and power laws of logarithms.
Achievement Indicators:
Develop and generalize the laws for logarithms, using numeric examples and exponent laws. Derive each law of logarithms
Determine, using the laws of logarithms, an equivalent expression for a logarithmic expression. Determine, with technology, the approximate value of a logarithmic expression, such as
Specific Outcome: Students will graph and analyze exponential and logarithmic functions.
Achievement Indicators:
Sketch, with or without technology, a graph of an exponential function of the form . Identify the characteristics of the graph of an exponential function of the form ,
including the domain, range, horizontal asymptote and intercepts, and explain the significance of the horizontal asymptote.
Sketch the graph of an exponential function by applying a set of transformations to the graph of , and state the characteristics of the graph.
Sketch, with or without technology, the graph of a logarithmic function of the form .
Identify the characteristics of the graph of a logarithmic function of the form , including the domain, range, vertical asymptote and intercepts, and explain the significance of the vertical asymptote.
Sketch the graph of a logarithmic function by applying a set of transformations to the graph of , and state the characteristics of the graph.
Specific Outcome: Students will solve problems that involve exponential and logarithmic functions.
Achievement Indicators:
Determine the solution of an exponential equation in which the bases are powers of one another. Determine the solution of an exponential equation in which the bases are not powers of one
another, using a variety of strategies.
Determine the solution of a logarithmic equation, and verify the solution.
Explain why a value obtained in solving a logarithmic equation may be extraneous. Solve a problem that involves exponential growth or decay.
Solve a problem that involves the application of exponential equations to loans, mortgages and investments.
Lesson 1: Exponential Functions
Problem 1: The graphs of the functions and are
shown on the right. Use a table of values to sketch the
graph of and on the same grid and then
label each graph.
Problem 2: Analyze these functions by stating the domain, range, x-intercepts, y-intercepts, and equations of any asymptotes. Describe what happens to the graph of as b gets larger.
An EXPONENTIAL FUNCTION is one whose equation is in the form
This function exhibits exponential growth (goes up to the right) if .
Problem 3: Use the graph of to sketch the graph of , then determine:
whether the function is increasing or decreasing
the intercepts
the equation of the asymptote
the domain and range of the function
Problem 4: Describe the transformations that will occur when the graph of is transformed into the graph of . What effect will this have on the graph?
Problem 5: The graph of , where b > 1, is translated such that the equation of the new graph is expressed as . Determine the range of this new function.
Problem 6: Could the table of values below describe an exponential function? Justify your answer.
x y -2 25 -1 5 0 1 1 0.2 2 0.04
Exponential graphs in the real world:
Assignment: Page 349 #1-13, MC #1,2
Lesson 2: Logarithmic Functions
Investigation: How are these graphs related?
Problem 1: The exponential function is shown below. Draw the inverse on the same grid. What are the equations for the asymptotes for the graphs? State the domain and range of each function.
In General: A logarithmic function is the inverse of an exponential function, , given by the equation:
.
Recall: Inverse means to switch x and y or reflect about the y = x line.
y intercept: none x intercept: 1
Dx: x > 0 Ry: y Є R
10
xy
10
log
y
x
The term logarithm is used to describe the inverse of a power. For example, the inverse of is the logarithm to the base 10 of x, which is written as . We say log base 10 of x.
Here is the graph of and its inverse.
Problem 2:Write each expression in exponential form:
a) b) c)
Problem 3:Write each expression in logarithmic form:
a) b) c)
Problem 4: Use benchmarks to estimate the value of each logarithm to the nearest tenth.
a) b) c)
x y Points
1 (1,0)
10 (10,1)
100 (100,2)
Problem 5: Graph . Identify the intercepts, the equations of any asymptotes, and the domain and range of the function.
Problem 6: Describe the transformations if the function is transformed into the function . Then analyze this function by stating the intercepts, domain, range, and asymptotes.
Lesson 3: Evaluating Exponents
Exponents
Problem 1: Simplify the following using exponent laws.
a) b) c)
Problem 2: Determine the exact value of the following powers without the use of a calculator. Decimals are NOT allowed!!
a)
b) c) d)
Problem 3: Rewrite the following in terms of the base given.
In General: If the bases of the powers in an equation are equal then the exponents must be equal.
Problem 4: Solving the following exponential equations using exponent laws.
a) b) c)
d) e)
Problem 5: Algebraically solve and verify the following radical equations.
a) b)
Lesson 4: The Laws of Logarithms
Property #1: The logarithmic function is the inverse of the exponential function
Property #2: The logarithm of a negative number is undefined i.e. is defined only if x > 0.
Property #3: The logarithm of a power with the same base is equal to the exponent
Property #4: You can change the base on the logarithm
Recall: Evaluate the following:
a) b) c) d)
e) f) g) h)
Problem 1:Rewrite the following in the form
a) b) c)
Problem 2:Rewrite the following as a single logarithm, and evaluate.
a) b) c)
The Power Law
The Product Law
Problem 3:Express the following as a single logarithm.
a) b)
Problem 4: Rewrite the following as a single logarithm.
Problem 5:If , then express the following in terms of x and y.
a) b)
Example 6:Write the expression in terms of log a and log b.
Type I Common Base: To solve an exponential equation, find a common base.
Type II Log both sides: Take the log base 10 of both sides to move the exponents down and then solve the equation. Round answers to the nearest hundredth.
Problem 1: Solve and verify the following exponential equations. Round answers to the nearest hundredth.
c) d)
e)
Assignment: Page 423 #7,8,10,12,14, MC #1
Recall: To change a number into a logarithm of any base
Logarithmic equations can be solved by using the following:
Write all terms of the equation as logarithms with a common base including constants.
Write each side of the equation as a single logarithm using the properties of logarithms
Use the properties below to eliminate the logarithms
IF , THEN A = B
OR IF , THEN
Problem 1:Solve the following logarithmic equations.
a) b)
Try 1: Solve the following logarithmic equations.
Problem 2:Solve and check the following logarithmic equations.
a) b)
c) d)
Lesson 7: Exponential and Logarithmic Applications Part I
Compound Interest and Population Growth
The formula may be used to solve many problems which involve exponential growth or decay, where A = amount after n time periods, = initial amount, i = where i is the percent increase (positive) or decrease (negative).
Problem 1: Barbara invests $8 000 in an account which pays interest at a rate of 6%/a compounded semi-annually.
a) How much money will Barbara have after 5 years?
b) How long will it take for her investment to triple?
Investments and Mortgages
When a series of equal investments is made at equal time intervals, and the compounding period for the interest is equal to the time interval for the investments, the amount in dollars, or future value FV, of these investments can be determined using this formula:
Problem 2: Determine how many monthly investments of $200 would have to be made into an account that pays 6% annual interest, compounded monthly, for the future value to be $100 000.
For mortgages, the formula above is adjusted slightly. The amount borrowed is called the present value PV, of the loan, R is the regular payment, n is the number of payments, and i is the interest per
compounding period.
Problem 3: A person want to borrow $200 000 as a mortgage to buy a house. The person can afford to pay $1500 a month. The mortgage will be repaid with equal monthly payments at 4% annual interest, compounded monthly. How many monthly payments will the person make?
where R dollars is the regular investment,
i is the interest per compounding period, and
Doubling Time and Half-Life
The formula may be used to solve many problems which involve doubling, tripling ... etc. or half-life, where a = initial amount, b = growth/decay factor, and p = growth/decay period.
Investigation: Finding the half-life of caffeine
http://www.energyfiend.com/the-half-life-of-caffeine
Problem 4: A sample of saliva was taken from a patient, and it was determined that 387 cells of bacteria were present. If this particular strain of bacteria doubles every 5 hours:
a) Determine the number of cells that will be present after 24 hours.
b) Determine the length of time it will take for this sample to grow to 5000 cells.
Lesson 8: Applications Part II
The Richter Scale
The magnitude of an earthquake, M, is given by the formula , where I is the earthquake
intensity and I0 is a reference intensity.
Problem 1: An earthquake in Turkey in 1999 had a magnitude of 7.4 on the Richter scale. An earthquake in Seattle in 1996 measured 5.3 on the Richter scale. How many times as intense was the earthquake in Turkey as the one in Seattle?
Problem 2: A major earthquake of magnitude 7.5 is 375 times as intense as a minor earthquake. Find the magnitude, to the nearest tenth, of the minor earthquake.
Loudness of Sound
The loudness of a sound was originally measured in Bels, named after Alexander Graham Bell. The current unit used is the decibel (dB), which is equal to one tenth of a Bel. The Bel scale, like the Richter scale, is logarithmic – a difference of 1 Bel, or 10 decibels, corresponds to a factor of ten difference in sound intensity.
The pH Scale
The pH scale measures the range of hydrogen ion concentration by determining the acidity or alkalinity of a solution. The scale measures from 0 to 14 with values below 7 representing increasing acidity, and values above 7 representing increasing alkalinity. The value of 7 represents the neutral level on the pH scale where the solution is neither acidic nor alkaline. Like the Richter scale, each increment on the pH scale is tenfold.
Problem 4: Acid rain has a pH of 4.2. How many times as acidic is acid rain as compared to water whose pH is 7?
Problem 5: Eggs have a pH of 8 and blood has a pH of 7.5. Eggs are how many times as alkaline as blood?
