Definition of a Logarithm

If b and N are positive numbers, b, and bx = N, then logb N = x.

Because any positive number, except 1, can be the base of an exponential expression, logarithms can have any positive number base except 1. There are an infinite number of logarithm bases possible. Only 2 bases are found on a hand- held calculator –common logarithms and natural logarithms

Common Logarithms

Common Logarithms are base 10 meaning that they are exponents of 10. There are many examples of common logs (short for logarithms) in the real world.

Richter Scale

An earthquake is the sudden release of energy caused by rock suddenly moving along an edge of the earth’s crust, called a fault. The release of energy is measured in joules. The range of joules can be from a very small number to millions of times that number. The range of numbers is too large to put in a scale, so the energy is expressed in scientific notation and only the exponents are used in the scale so that sequential numbers can be used.

Richter

magnitudes / Description
1 / Cannot be felt except by instruments
2 / Cannot be felt except by instruments
3 / Cannot be felt except by instruments
4 / Like vibrations from a passing train
5 / Strong enough to wake sleepers
6 / Very strong; walls crack, people injured
7 / Ruinous; ground cracks, houses collapse
8 / Very disastrous; few buildings survive, landslides

An increase of 1 on the Richter scale corresponds to a multiplication of energy released by a factor of 10. An earthquake with Richter magnitude 5.6 is ten times as strong as an earthquake of magnitude 4.6.

In order to compare the strength of two earthquakes, subtract the Richter magnitudes and raise 10 to that power.

ExampleAn earthquake in Alaska in 1964 had a Richter magnitude of 8.6. The San Francisco earthquake of 1906 had an estimated magnitude of 8.3. How much stronger was the Alaska earthquake than the San Francisco earthquake?

Solution8.6 – 8.3 = 0.3 100.3. The Alaska earthquake was approximately twice as strong as the San Francisco earthquake.

1. The strongest Richter magnitude ever recorded was 8.9. This reading occurred twice, once in the Pacific Ocean near the Colombia-Ecuador border in 1906 and once again in Japan in 1933. How much stronger were those earthquakes than the San Francisco earthquake?

2.In 1985 Mexico City recorded an 8.1 magnitude earthquake. How much stronger was the Alaska earthquake?

Decibel Scale

The intensity of sound is measured by the Decibel scale. The decibel is 0.1 of a bel, a unit named in honor of Alexander Graham Bell, the inventor of the telephone. It also is a logarithmic scale. Sound is measured in watts/square meter. The softest sound that a human can hear is 10-12 w/m2. The noise of a jet plane is 102 w/m2. This also is a very large range, so only the exponents are used in the scale. Since deci means a tenth, the decibel readings are multiplied by 10.

The chart below gives the decibels for some common sounds.

Common Sound / Decibels
Jet plane (30m away)
Pain threshold
Amplified rock music (2m away)
Noisy kitchen
Heavy traffic
Normal conversation
Average home
Soft whisper
Threshold of sound / 140
130
120
110
100
90
80
70
60
50
40
30
20
10
0

In order to compare sound intensity, divide the decibels by 10 and then subtract. The answer is the exponent of 10.

Example: How much more intense is the noisein an average home than the threshold of sound (when the softest sound is first heard)?

Solution: The decibel level of the average home is 50 and the threshold of sound has a decibel level of 0. 5 – 0 = 5, so the intensity of the sound in the average home is 105 or 100,000 times as intense as the first sound that can be heard.

1.How much more intense is the sound of a jet plane than the sound of heavy traffic?

2.How much more intense is the sound of amplified rock music than the sound of normal conversation?

3.How much less intense is the sound of a soft whisper than normal conversation?

The pH scale is a logarithmic scale that is used to measure how acidic or alkaline a solution is. This is done by measuring the concentration of hydronium ions, H3O+, in the solution. The concentration is expressed as a power of 10. The exponent of 10 is the pH value. Pure water has a pH of 7. Acidic solutions have pH values less than 7 and alkaline or base solutions have pH values greater than 7.

1.The gastric juice in your digestive system has a pH of 2.0 and many soft drinks have a pH of 3. How much more acidic is gastric juice than a soft drink?

2.Seawater has a pH of 8.5. How much more alkaline is seawater than pure water?

Natural Logarithms

Natural logarithms are base e logs. As in the case of , e is a number and not a variable. It is used in honor of Leonard Euler, an 18th century Swiss mathematician. In order to understand the number e, we will revisit investments.

In our study of exponential functions, we used the formula A = P( 1 + r )t. This formula is for interest that is compounded once a year. Interest can be compounded more than once a year. In order to calculate the amount of money after a specified period of time, the formula must be expanded to allow compounding more than once a year. The formula

is used for more than annual compounding. The number of times per year that compounding is done is n. The number of years is t.

Example: You invest $1000 at 5% annual interest, compounded quarterly for 4 years. How much money will you have?

Solution

1.How much money would you have after 3 years if you invested $500 at 7% annual interest, compounded monthly?

2.How much money would you have if you invested $2500 at 5.75% annual interest, compounded daily for 5 years?

A theoretical investment will lead us to an understanding of the number e. Suppose you invest $1 at 100% interest for one year. Fill in the table below for different values of n.

For this investigation do not round. Use all the decimal places displayed on your calculator.

Compounded / Value of n / Process / Amount
Annually / 1 / / $2.00
Semi-annually / 2 / / $2.25
Quarterly
Monthly
Daily
Hourly
Every minute
Every second

The sequence of values for the total amount gets closer and closer to e. To display e on your calculator, press 2nd LN. Above the LN key you will find ex; therefore this key sequence will raise e to the power you enter. Press 1 to raise e to the first power and ENTER.

e =

In a more realistic situation, suppose a bank pays 5% interest on $1 for one year. Find the amounts earned under various compounding methods.

Word / Values of n / Process / Amount
Annually / 1 / / $1.05
Semi-annually / 2 / / $1.050625
Quarterly
Monthly
Daily
Hourly
By the minute
By the second

The total amount gets closer and closer to $1.051271096… This is the value of $e0.05. Find e0.05 on your calculator. For situations where interest is compounded continuously, the formula for compound interest must be changed because n cannot be defined for a continuous process. The formula for continuous compounding is actually much simpler than the general compounding formula.

For continuous compounding use the formula .

1.How much would you have if you invested $500 at 7% annual interest compounded continuously for 3 years? How does that compare to the same investment you calculated earlier?

2.How much would you have if you invested $2500 at 5.75% annual interest compounded continuously for 5 years? How does that compare to the same investment you calculated earlier?

SATEC/Algebra II/Logarithmic/7.04 Definition of Logarithms/Rev. 7-01Page 1 of 8