Logarithm Sp 99

How to Handle Significant Figures in Logarithm Dr. Yau

Outline of How to Handle Sig.Fig. in Logarithm
If you want a complete review, click on each link below in order.

Brief Review of Logarithm
Logarithm of Numbers with Exponents that are Integers
Antilog_of_Integers

Logarithm of Numbers with Exponents that are Non-integers
SIGNIFICANT FIGURES When Finding the Log of a Number
Determining the Antilog of Numbers that are Non-integers
SIGNIFICANT FIGURES When Finding the Antilog of a Number
How to Solve Simple Equations Involving Logarithms
SUMMARY
APPLICATIONS TO CHEMISTRY PROBLEMS

Brief Review of Logarithm

Logarithm is the exponent to which a number is raised to produce a given number.

The common logarithm, commonly abbreviated as log, refers to the exponent to which 10 has to be raised, and the natural logarithm, abbreviated ln, refers to the exponent to which e has to be raised. We will consider only the common logarithm here.

· For example, given the number 100, log of 100 is 2 since 10 has to be raised to a power of 2 to produce 100.

· In the case of log 0.01, the answer is – 2 since 0.01 = 10 ^–2 and we must raise 10 to the power of –2to produce 0.01.

When we want to find the log of a number, we are asking what is x in 10^x to produce this number. See below for more examples.

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Logarithm of Numbers with Exponents that are Integers

Log of numbers with exponents that are integers (i.e., numbers that are simple powers of ten) do not require the use of calculators or log tables to compute. First express the number in exponential form, and the log of the number is simply the exponent.

See examples shown in the table below:

log 100	= log 10²	= 2
log 10	= log 10 ¹	= 1
log 1	= log 10⁰	= 0
log 0.1	= log 10^{– 1}	= –1
log 0.01	= log 10^{– 2}	= –2
log 0.00001	= log 10 ^–5	= –5

Note that when a number increases by powers of ten, such as from 10 to 100, the logarithm of these numbers increase only by one unit (from 1 to 2). This is referred to as a "logarithmic function". Certain measurements in chemistry are related to each other by such logarithmic functions (e.g. % transmittance vs. absorbance, H₃O⁺ vs. pH).
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Antilog of Integers

Antilog refers to the inverse function of log. It can also be written as log^{– 1 .}

To find the antilog of a number, we simply reverse the process. It is the value equivalent to 10 brought to the power of the given number. In other words, antilog of 2 is 10² or 100. The antilog of –2 is 10^–2 or 0.01. Again, this does not require a calculator or a log table to compute.

See examples shown in the table below:

antilog 2	= 10 ²	= 100
antilog 1	= 10 ¹	= 10
antilog 0	= 10 ⁰	= 1
antilog – 1	= 10 ^{– 1}	= 0.1
antilog – 5	= 10 ^{– 5}	= 0.00001

Since the antilog function is the inverse of the log function, the antilog of the log of a number is the number itself. For example, antilog log 100 = 100.
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Logarithm of Numbers with Exponents that are Non-integers

Logarithm of more complex numbers require the use of a calculator or a log table. Only the use of a calculator will be discussed here.

If you had to calculate the log of a number, just enter the number and then press log.

For example, to find log 2.5, enter 2.5 then press log. The answer is 0.397940009.

For certain calculators, such as graphics calculators, you need to press log first, followed by the open parenthesis sign ( , the number, the close parenthesis sign ) , and finally enter . For such calculators, it is wise to get in the habit of always putting the number within the parenthesis signs. In other words, to find log 2.5, it is best to press log ( 2.5 ) enter . Although it makes no difference in this case, it can cause trouble in other situations if the parentheses were omitted. See below for example:

If you want to find log3x5, which has the answer 1.17609…, and you pressed log 3 x 5 enter , you would have computed (log 3) x 5 instead (giving you the answer of 2.3856…).

Especially if you are one of those who insist on entering numbers like 2x10³ by pressing 2 x 10^x3 instead of the recommended keystrokes of 2 EE 3, you will have to think very carefully about the order of operation. The calculator will do the log function before multiplication and division. So, if you want to find log 2x10³ (with the answer of 3.3010…) and pressed log 2 x 10^x3 enter , you would have computed for (log 2) x 10³instead (giving you the answer of 301.0299…) The problem is avoided by enclosing your number within the parentheses signs.

Practice by finding the log of the numbers in the table below:

log 4.2	0.62324929…
log 4.2x10²	2.62324929…
log 4.2x10⁵	5.62324929…
log 4.2x10^–1	–0.37675071…
log 4.2x10^–2	–1.37675071…

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Significant Figures When Finding the Log of a Number

Before we begin, let us establish the vocabulary involving exponential expressions and logarithms:

In an exponential expression such as 4.2 x 10 ^{– 2 ,} "4.2" is called the coefficient, and – 2 is called the exponent. Thus, 2 x 10³ would have a coefficient of 2 and an exponent of 3.

The log of a number consists of two parts. The number to the left of the decimal point is called the characteristic, and the number to the right is the mantissa. Thus, the number 3.278 has a characteristic of 3 and a mantissa of 278. The number – 3.278 has a characteristic of – 3.

Let us examine the table again:

log 4.2	0.62324929…
log 4.2x10²	2.62324929…
log 4.2x10⁵	5.62324929…
log 4.2x10^–1	–0.37675071…
log 4.2x10^–2	–1.37675071…

As you can see in the first three examples in the table, the mantissa is established by the coefficient (4.2) and the characteristic is established by the exponent. In the last two examples, this may not be as apparent,
                          but log 4.2x10^–1
is calculated as        log 4.2      +   log 10^–1
which is equal to     0.6232… +   – 1
                             = – 0.37675071…

The point is, the significant figures of a number should be reflected only in the mantissa of its log! In other words, the number of significant figures of a number is the number of decimal places in its log.

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
CAUTION:
To figure out the number of decimal places, the number MUST be first expressed in decimal form, rather than in exponential form!!!
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For example, 4.2 has 2 sig. fig., so log 4.2 should have only 2 decimal places: 0.62

4.25 has 3 sig. fig., so log 4.25 should have 3 decimal places: 0.628
0.03 has 1 sig. fig., so log 0.03 should have 1 decimal place: -1.5
1.01 has 3 sig. fig., so log 1.01 should have 3 decimal places: 4x10^-3 = 0.004

Practice with the examples below:

log 4.2	0.62
log 4.2x10²	2.62
log 4.2x10⁵	5.62
log 4.2x10^–1	–0.37
log 0.002735	–2.5630
log 3200	3.50
log 50000	4.7
log 1.002	9x10^-4

If you had trouble with the last 4 examples, you need to review the rules for significant figures.
Click here to review the rules for significant figures.

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Determining the Antilog of Numbers that are Non-integers.

Again, either a calculator or a log table is required. Only the use of a calculator will be discussed here.

If you have to find the antilog of 3.5, simply enter 3.5 then press the 10^x key. Generally the log and the 10^x functions share the same key, but involves pressing either the 2^ndkey, the SHIFT key or the INVERSE key first.

With the graphics calculator, you would press the 10^x key first before entering the number. Again, it would be wise to always make use of the parenthesis signs.

Practice with the following examples:

antilog 3.5	3162.27766…
antilog 0.35	2.238721139…
antilog – 3.5	3.16227…x10⁴

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Significant Figures When Finding the Antilog of a Number

You just need to reverse your thinking. The number of decimal places in the number determines the number of significant figures in it antilog. Start by examining the table under the heading "Significant Figures When Finding the Log of a Number", and note that the number of decimal places in the second column corresponds to the number of significant figures in the left column.

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
CAUTION:
To figure out the number of decimal places, remember to convert the number into decimal form first, rather than counting the decimal places while it is still in exponential form. For example, 2.15 x 10^–1 has 3 decimal places not 2, since we are really considering 0.215.
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Now practice with the examples below:

antilog 2.78	6.0x10²
antilog 12.275	1.88x10¹²
antilog – 3.8	2x10^–4
antilog 15	10¹⁵
antilog 1.23x10^–2	1.029
antilog 3x10^–4	1.001
antilog 9.6x10^–3	1.022

If you had trouble with the last 3 examples, you probably forgot that the numbers in the first column have to be converted into decimal form before you count the number of decimal places. Thus, 1.23 x 10^–2is equivalent to 0.0123, which has 4 decimal places, and its antilog should have 4 significant figures (1.029).

In the case of the antilog of 15 (which has no decimal places), the answer has no significant figures, as indicated by merely 10¹⁵.

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How to Solve Simple Equations Involving Logarithms

Given the equation log x = 7.82, you would first find the antilog of both sides of the equation:
                   antilog log x = antilog 7.82
                                     x = antilog 7.82
                                        = 6.606934…x10⁷
                                        = 6.6x10⁷ (Ans. has 2 sig.fig.)

Given the equation log 5.42 x 10^–2 = x, you would simply find the log of the number:
– 1.266 = x. (Ans. has 3 decimal places)

Practice with the following examples:

log x = 4.28 x 10^–2	x = 1.104 (Remember to find x you need to find the antilog of 4.28x10^{-2 ,} which is 0.0428 and has 4 decimal places. Ans. should have 4 sig.fig.)
log x = 0.14 x 10^–1	x = 1.03 (3 sig.fig. because 0.14x10^-1 has 3 decimal places.)
log 0.053 = x	x = 1.28 (2 decimal places)
log 10⁴^.3 = x	x = 4.3 (review definition of log)

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Summary

To find the log of a number:

1. Enter the number then press LOG.
For graphics calculators, press LOG ( the number ) ENTER

2. Express the answer in decimal form.

3. The number of sig. fig. in the original number is the number of decimal places in the answer.

Quick check: log 7.6x10^–3 = – 2.119186 = – 2.12

To find the antilog of a number:

1. Enter the number then press 10^x .
For graphics calculators, press 10 ^x ( the number ) ENTER .

2. Convert the original number to decimal form and count the number of decimal places.

3. The number of decimal places is the number of sig. fig. in the answer.

Quick check: if log x = – 6.2x10^–1

then x = antilog – 6.2x10^–1 = 2.398832x10^–1 = 2.4x10^–1

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APPLICATIONS TO CHEMISTRY PROBLEMS

1. Percent transmittance (%T) is related to the absorbance (A) by the equation: A = - log (%T/100).

a) Calculate A if %T is 72.9.