A LOGARITHM is an exponent



A LOGARITHM is an exponent.  It is the exponent to which the base must be raised to produce a given number. For example, what is the answer here? 23  =  _, then 3 is called the logarithm of _ with base 2. We could also write this 3  =  log2_, where 3 is the exponent to which 2 must be raised to produce _. We write the base as a subscript.

Since 104 = 10,000 then log1010,000 = 4 - "The logarithm of 10,000 with base 10 is 4." 4 is the exponent to which 10 must be raised to produce 10,000. - "104 = 10,000" is called the exponential form. "log1010,000 = 4" is called the logarithmic form.

Here is the definition:

logbx = n   means   bn = x.

That base with that exponent produces x.

Indexes and Bases

Consider a number written as 32. The number 3 is called the base; the number 2 is the index (plural indices). 32 is read as three squared and means multiply 3 by itself twice. In other words 3 × 3 = 9.

Other examples:

23 (read as 2 cubed) = 2 × 2 × 2 = 8

34 (3 to the power of 4) = 3 × 3 × 3 × 3 = 81

There are four simple laws of indices that apply to numbers with the same base:

|am × an = a(m + n) |(When two numbers are multiplied, their indices are added) |

|am / an = a(m - n) |(When two numbers are divided, their indices are subtracted) |

|n√am = am / n |(For the nth root divide the index by n) |

|(am)n = am × n |(Raising a number to the nth power, multiply the index by n) |

Here are some examples of these laws:

23 × 24 = (2 × 2 × 2) × (2 × 2 × 2 × 2) = 2(3 + 4) = 27 = 128 (Adding indices)

35 / 32 = (3 × 3 × 3 × 3 × 3) / (3 × 3) = 3(5 - 2) = 33 = 27 (Subtracting indices)

√44 = 4(4 / 2) = 42 = 16 (Dividing indices)

(23)2 = 2(3 × 2) = 26 = 64 (Multiplying indices)

Note that indices make calculations simpler. These indices can be used to define Logarithms.

We can say that:

3 is the logarithm of 8 to base 2 (because 23 = 8)

and

4 is the logarithm of 81 to base 3 (because 34 = 81).

These are written: log28 = 3 and log381 = 4

Common logarithms

The system of common logarithms has 10 as its base.  When the base is not indicated, log 100 = 2

Then the system of common logarithms -- base 10 -- is implied.

Here are the powers of 10 and their logarithms:

|Powers of 10:   |  |

|Logarithms: |  |−3 |

|  | | |

"The logarithm of a quotient is equal to the logarithm of the numerator

minus the logarithm of the denominator."

3.    logb x n  =  n logbx

"The logarithm of a power of x is equal to the exponent of that power

times the logarithm of x."

Application to pH problems:

pH = -log (hydrogen ion concentration) = -log [H+]

• Example: What is the pH of an aqueous solution when the concentration of hydrogen ion is 5.0 x 10-4 M? pH = -log [H+] = -log (5.0 x 10-4) = - (-3.30) = 3.30

pH = -log (hydrogen ion concentration) = -log [H+]

• Example: What is the concentration of the hydrogen ion concentration in an aqueous solution with pH = 13.22?

pH = -log [H+] = 13.22

log [H+] = -13.22

[H+] = inv log (-13.22)

[H+] = 6.0 x 10-14 M

Logarithmic calculations:

|Common Logarithm |

|log xy = log x + log y |

|log x/y = log x - log y |

|log xy = y log x |

QUIZ: Give the correct answer to the correct number of significant figures.

|Question 1 |log 23.0 = ? |

|Question 2 |What is the pH of an aqueous solution in which [H+] = 2.7 x 10-3 M? |

|Question 3 |Find x if log x = 10.23. |

|Question 4 |If the pH of an aqueous solution is 6.52, what is the concentration of the hydrogen ion? |

|Question 5 |log (1.2 x 106)3 = ? |

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Answers: (1) 1.362 (2) 2.57 (3) 1.7 x 1010 (4) 3.0 x 10-7 M (5) 18.24

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IV. LOGARITHMIC FUNCTIONS

        The logarithmic function is the inverse of the exponential function.  In the exponential function, y = ax, we say that a is the base.  Now the inverse function is written x = loga y.  This function is called the logarithm of y to the base a.  Therefore, the logarithm of a number (y) is the power (x) to which the base must be raised to equal that number. y = ax     x = loga y

        Although a can be any positive, real number except 1, only two bases are commonly used.  These are: base 10 (a = 10), called the common logarithm (or log10) and base e (a = e = 2.718...), called the natural logarithm (loge or ln). In this module, we will work with common logarithms (log10). 

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V. COMMON LOGARITHMS

        A common logarithm of a number is the power to which 10 must be raised to equal that number: log10 y = x or 10x = y.  When dealing with common logarithms, the expression log10 is usually simplified to log.  For example:

• the log of 100 = 2 (because 102 = 100)

• the log of 1000 = (103) = 3

• the log of 158 = (102.2) = 2.2

• the log of 1 = 0 (because 1 = 100)

• the log of 0.01 = -2 (because of 10-2 = 0.01)

• the log of 0.0001 (10-4) = -4

• the log of 0.05012 (10-1.3) = -1.3

        From these examples, you can see that the logarithm of a positive, real number can be positive or negative or zero.

• If y (the positive, real number) is greater than 0 but less than 1, log y will be negative.

• If y = 1, log y = 0

• If y is greater than 1, log y will be positive.

The log of a number can be found using a slide rule or from log tables.  However, most modern calculators can calculate logs.   Before we do this, however, let's review how logarithms of numbers were determined before calculators were readily available so that we can predict the approximate magnitude of a log.

        The first step in determining a logarithm of a number is to write the number in exponential form as a coefficient (the value ranging from 1.000 to 9.9999...) and ten raised to some power.  Suppose we wish to find the log of 600.  We first write this as 6.00 x 102 (assuming the zeros are significant).  Log (6.00 x 102) equals log (6.00) + log (102).  You know that log (102) equals 2.   You know that log of a number greater than 6 is positive.  (Actually, it will be 2.778.)

        Suppose we wanted to find the log of the number 0.0600.   First, we rewrite this in exponential form:  6.00 x 10-2.   Then we can write this as log (6.00) + log (10-2).  Log (10-2) is -2.  Log (6.00) will be positive (0.778).  Therefore the log of 6.00 x 10-2 will lie between -1 and -2.  Actually, the log of 6.00 x 10-2 is 0.778 + (-2) or -1.222.  By looking at the number (written in exponential form), you should be able to predict the magnitude of the log. It is generally very easy to use your calculator to find a logarithm.  Most calculators use similar buttons to do logs. 

|Sample Problem 3:  Find the log of 0.02500. |

| Solution:  The log of 0.02500 is -1.6021. |

|Explanation: The number as 2.500 x 10-2, we predicted that the log should lie between -2 and -1.  The number has four significant figures; therefore the |

|mantissa must have four significant figures:  -1.6021. |

|Sample Problem 2:  Find the log of 3.0 x 10-9. |

|Solution:  The log of (3.0 x 10-9) is -8.52. |

| |

|Explanation: We predicted the number would like between -9 and -8.  The number had two significant figures; therefore the mantissa contains two significant |

|figures (-8.52). |

|Sample Problem 1:  Find the log of 3 x 109. |

|Solution:  The log of (3 x 109) is 9.5.  (We predicted that the log would lie between 9 and 10.) |

To do logs on the calculator, follow these steps:

Step 1:  Enter the number Step 2:  Push the [LOG] button

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       Significant Figures and Logarithms:   In Sample Problem 1 above, the answer was written as 9.5.  The number to the left of the decimal is called the characteristic.  It represents the exponent.  In this example, the number was raised to the 9th power.  In this number, 9 is the characteristic.  The number to the right of the decimal place is called the mantissa. In this example, 5 is the mantissa.

        When determining the number of significant figures to use when reporting logarithms, only the number of digits in the mantissa are significant.  The characteristic (to the left of the decimal place) represents the magnitude of the number. The mantissa should have the same number of significant figures as in the original number.  The number 3 x 109 has 1 significant figure.  Therefore, the log should have 1 decimal place. In the number  9.5, only the 5 is significant.]

VII. CALCULATING pH

        Logarithms are used extensively in the area of acid-base chemistry and biology.  You will frequently have to calculate the pH of a solution from the hydrogen ion concentration [H+].  This involves taking a logarithm. pH is defined as the negative log of the hydrogen ion concentration.  (Actually, the "p" means "the negative log".   Thus pAg would be -log [Ag] and pCu would be -log[Cu].  You get the idea.) The pH function is represented mathematically as:

pH = -log [H+] or, alternatively,  pH = log 1/[H+]

Calculating the pH of a solution involves taking the log of the hydrogen ion concentration, then changing the sign.  For example, to calculate the pH of a solution that is 2.50 x 10-5 M in hydrogen ion ([H+]):

1. Enter the value (2.50 x 10-5) into your calculator

2. Push [LOG]

3. The display is -4.602

4. Since pH = -log [H+], pH = -(-4.602) or pH = 4.602

        Alternately, you could use the definition pH = log 1/[H+]:

1. Enter the number (2.50 x 10-5) on your calculator and push [1/x]

2. The display is 40000

3. Push [LOG]; the answer is 4.602

| |

|Sample Problem 5:  Calculate the pH of a solution that is 4.81 x 10-2 M in hydrogen ion ([H+]). |

|Solutions: |

| |

|pH = -log [H+] |

|pH = -log (4.81 x 10-2) |

|pH = -(-1.318) |

|pH = 1.318 |

|or |

|pH = log 1/[H+] |

|pH = log 1/(4.81 x 10-2) |

|pH = log 20.79 |

|pH = 1.318 |

| |

 

VIII. CALCULATING [H+] FROM THE pH

Alternately, you may know the pH of a solution and want to find the hydrogen ion concentration. This involves the antilogarithm function. Since pH = -log [H+], we can take the antilog of both sides to yield 10-pH = [H+]. This can be represented mathematically as:

[H+] = 10-pH or  [H+] = antilog (-pH)

        To solve:  Enter [-pH], then take the antilog.

| |

|Sample Problem 6:  Find the hydrogen ion concentration in a solution whose pH = 7.35 |

|Solution:  Take the antilogs of both sides of the equation:  [H+] = antilog (-7.35) = 10-7.35 = 4.5 x 10-8 |

| |

|Explanation:  Because there were two decimal places in the log, the concentration should have two significant figures. |

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