Logarithm significant figures

For base-10 logarithms, the rules governing significant figures are as follows ... 

In taking the inverse logarithm of a number, retain as many significant figures as there are after the decimal point in the number. Thus... 

The rules for significant figures involving natural logarithms and inverse logarithms are somewhat more complex than those for base-10 logs. However, for simplicity we will assume that the rules listed above apply here as well. Thus... 

Significant figure A meaningful digit in a measured quantity, 9,20-2 lq ambiguity in, 10 in inverse logarithms, 645-647 in logarithms, 645-647 Silicate lattices, 243 Silicon, 242-243 Silver, 540-541 Silver chloride, 433,443-444 Simple cubic cell (SC) A unit cell in which there are atoms at each comer of a cube, 246... 

A pH around 3 represents an acidic solution, so we expect the hydronium ion concentration to be much larger than the hydroxide ion concentration. Remember that although pH and. w are dimensionless, concentrations of solutes always are expressed In mol/L. Our results have two significant figures because the logarithm has two decimal places. 

Note To find e-95 66, take the inverse In of-95.66 on your calculator, inv In of-95.66 = 2.85 x 10 42. Keep one more significant figure and round off to three significant figures at the end, particularly when working with logarithms. 

Note Keep all the significant figures and round at the end. Remember the number of decimal places in pH or pOH values are set by the number of significant figures in the [H+] or [OH-] this is a result of working with logarithms. 

Note Significant figures for logarithms is equal to the number of significant figures in the mantissa. 

In the conversion of a logarithm into its antilogarithm, the number of significant figures in the antilogarithm should equal the number of digits in the mantissa. Thus,... 

Significant figures in logarithms were discussed in Section 3-2. 

In nearly all practical chemical calculations, a precision of only two to four significant figures is required. Therefore the student need not perform multiplications and divisions manually. Even if an electronic calculator is not available, an inexpensive 10-in slide rule is accurate to three significant figures, and a table of 4-place logarithms is accurate to four significant figures. 

A couple of comments are in order here. First, did you notice the value of the pH is the same as the absolute value of the exponent This will always be true when the first part of the scientific notation is exactly 1. The second comment relates to significant figures. There are two significant figures in the molarity measurement of 1.0 x 10 3 M. There are also two significant figures in the pH value of 3.00. Finally, pH values have no intrinsic units. Logarithms represent pure numbers, and as such, have no units. 

In this unit you will find explanations, examples, and practice dealing with the calculations encountered in the chemistry discussed in this book. The types of calculations included here involve conversion factors, metric use, algebraic manipulations, scientific notation, and significant figures. This unit can be used by itself or be incorporated for assistance with individual units. Unless otherwise noted, all answers are rounded to the hundredth place. The calculator used here is a Casio FX-260. Any calculator that has a log (logarithm) key and an exp (exponent) key is sufficient for these chemical calculations. 

At this point we need to discuss significant figures for logarithms. The rule is that the number of decimal places in the log is equal to the number of significant figures in the original number. Thus... 

Use a calculator to find a number that is equal to the reciprocal of its own natural logarithm. Report the answer to four significant figures. 

In a logarithm of a number, keep as many digits to the right of the decimal point as there are significant figures in the original number. 

The number to the left of the decimal point in a logarithm is called the characteristic, and the number to the right of the decimal point is called the mantissa. The characteristic only locates the decimal point of the number, so it is usually not included when counting significant figures. The mantissa has as many significant figures as the number whose log was found. 

In logarithms, it is the mantissa that determines the number of significant figures. 

To solve this type of problem it is convenient (but not necessary) to sketch an Arrhenius plot this is shown in Figure 9.11. It is to be emphasized that in taking the logarithms of the rate constants, and the reciprocals of the rates, it is necessary to use enough significant figures, as we are dealing with relatively small dijfferences between the values. 

To represent pH to the appropriate number of significant figures, you need to know the following mle for logarithms ... 

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