By the end of this lesson, you should be able to:

2. Identify 10 as the base for common logs, and "log" as the standard function name, and manipulate common logs.

3. Identify e as the base for natural logs, and "ln" as the standard function name, and manipulate natural logs.

4. Use division to convert from any log base to any other log base.

5. Solve equations containing logarithmic expressions.

6. Solve equations where the argument of a logarithmic function is a product, quotient, or exponential expression.

7. Explain the concept of a logarithmic scale and cite at least one example.

8. Illustrate axis values for logarithmic and semi-logarithmic graphs.

Logarithms can be are very useful in solving problems and in presenting findings. This lesson introduces common and natural logarithms and explains the basic operations involved.

Logarithms

Logarithms deal with exponents: raising one number to the power of another number. Ten raised to the power of three is:

• 10³ = 10 * 10 * 10 = 1000

In this example, 3 is the exponent or power, and 10 is the base. (Please do not confuse this with another use of the word, "base," that is used to distinguish the binary system, the decimal or base 10 system, and the hexadecimal system.)

Answer the following:

• 10 raised to what power gives us 1000?

The answer is 3.
Using a base of 10, the logarithm of 1000 is 3.
log₁₀ 1000 = 3
Because 10³ = 1000

In general

• log_b x = y, where b^y = x

Bases and Types of Logarithms

Common Logarithms

The base of a logarithm can be any positive real number other than 1, but when the base is 10 the logarithm is called a common logarithm or a common log. A common abbreviation for this function on calculators and in programming languages is log(x). Unless another base is indicated, the base is assumed to be 10.

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There is a special number, a constant called e, that is used as a logarithmic base. That number is approximately equal to 2.71828. When it is used as the base, the logarithm is called a natural logarithm or a natural log. A common abbreviation for this function on calculators and in programming languages is ln(x). 

• ln(x) = log_e x
• If ln(x) = y, then log_e x = y, and e^y = x

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One of the great properties of logs is that they let you change bases by simple division.

• log_b x = log_a x / log_a b

This can be done with Natural logs as well, using the following equation:

• log_b x = ln(x) / ln(b)

Your calculator might let you find the value of log(45) by using the common log function. But you can also find it with the natural log function by typing in

• ln(45)/ln(10)

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1. log₁₀ 100,000 = ? Mouseover for the answer.
2. log₁₀ 0.0001 = ?  Mouseover for the answer.
3. log₂ 16 = ?  Mouseover for the answer.
4. log₅ 625 = ?  Mouseover for the answer.
5. log_b 64 = 3; b = ? Mouseover for the answer.
6. log₁₀ M = -2; M = ? Mouseover for the answer.
7. log₂₅ 5 = ?  Mouseover for the answer.
8. log 100 = ?  Mouseover for the answer.
9. log(25/250) = ?  Mouseover for the answer.
10. log(100) + log(1000) = ?  Mouseover for the answer.
11. ln(125) / ln(5) = ?  Mouseover for the answer.

Logarithmic Operations

Logarithms are exponents. Therefore, we can use the neat tricks associated with exponents. For example, when we multiply two numbers with the same base, we add their exponents.

• b^x b^y = b^(x+y)

Therefore, when we wish to find the log of a product, we can add the logs of the multiplicands.

• log_b (XY) =  log_b X + log_b Y

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Similarly, when we divide two numbers with the same base, we subtract one exponents from the other.

• b^x / b^y = b^(x-y)

Therefore, when we wish to find the log of a quotient, we can subtract the log of the divisor from the log of the dividend.

• log_b (X/Y) =  log_b X - log_b Y

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When raise a number to a power, and then raise that expression to another power, we multiply the exponents.

• (b^x)^y = b^(xy)

Therefore, since a logarithm is a type of exponent, if we wish to find the log of a number raised to a power, we can use that power as a coefficient: 

• log_b X^y =  ylog_b X

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Recall that the square root of a number is obtained by raising that number to the power of 1/2, the cubed root is obtained by raising a number to 1/3, and so on. 

• The n^th root of x is x^1/n 

We can also use logarithms to extract roots. Applying the last rule, but substituting a fractional exponent, we get:

• log_b X^m/n = (m/n) log_b X

Thus, the log of the square root of X would be:

• log_b X^1/2 = (1/2) log_b X
• the natural log of the square root of 9 would be: .5 ln(9), which is equal to ln (3)

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12. log (100 * 10,000) = ?  Mouseover for the answer.
13. log (10X) = ?  Mouseover for the answer.
14. ln x⁴ = ?  Mouseover for the answer.
15. log 10x - log x = ? Mouseover for the answer.
16. log₅125⁴ = ? Mouseover for the answer.

Logarithmic Scales and Graphs

Logarithmic Scales

One application of logarithms concerns the use of logarithmic scales to describe and display data.

In a standard, or arithmetic scale, every unit of the scale is equal to every other unit; if there is one inch between 100 and 200, then there is another inch between 200 and 300.

In a logarithmic scale, however, each inch is equal to the same percentage change. For example, if there is one inch between 10 and 100, then there is another inch between 100 and 1000. Sometimes, a chart or graph uses a logarithmic scale for only one of two axes, in which cases it is called a semi-logarithmic.

One example of a logarithmic scale is the decibel (dB) scale of sound level. For each 10 decibels, the intensity increases by a factor of 10. If sound X carries ten times the amount of energy as sound Y, then sound X is 1 bel, or 10 decibels greater.

For example, quiet conversion may have a sound intensity level of 60 dB, which corresponds to 10^-6 Watts per square meter. A typical factory may have background noise at 80 dB, which corresponds to 10^-4 Watts per square meter. 70 dB is ten times as intense as 60 dB; 80 dB is ten times as intense as 70 dB. The factory is one hundred times as loud as the quiet conversation.

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Logarithmic scales are also useful in charts and graphs, especially when the rate of change is geometric. Here is a comparison of data charted on arithmetic and logarithmic scales. (The charts are on age-specific cancer incident rates for Pennsylvania residents in 1988, and can be found at: http://www.health.state.pa.us/HPA/Stats/techassist/arithlog.htm ).

First, here is an example of an arithmetic scale:

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The third graph, below, is different data plotted on a graph that has a logarithmic scale for the x axis.

A semi-logarithmic chart from
http://www.tulane.edu/~sanelson/geol204/floodhaz.htm

In general, if a rate of increase is based on an exponential increase, such as doubling each day, the graph will result in a curve on a standard scale, and could be a straight line on a logarithmic scale.

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One more example, this one from:
http://www.clir.org/pubs/reports/conway2/index.html

References and Sample Problems

A basic introduction to logarithms, much like this one, can be found at:

http://www.nyctc.cuny.edu/science/safety/logarithms_intro.htm

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To see how logarithms can be used to help us solve problems, visit the following:

http://www.nyctc.cuny.edu/science/safety/logexamples.htm

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For more practice with logarithms, look at:

http://work.ccv.vsc.edu/pubs/mathchps/logs.pdf

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An introduction to log scales can be found at:

http://physics.mtsu.edu/~wmr/log_1.htm