what is the characteristics of 0.00023
Answers
What is the characteristic of log of 0.0054 and why?
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Answer: -3
Discussion:
By definition, the integral part of a logarithm is called the characteristic, and the fractional part when expressed as a positive decimal is called the mantissa. The logarithm of a number which has one digit before the decimal point is wholly fractional, that is the logarithm lies between 0 and 1.
While dealing with numbers, we work in the system of common logarithms where the base is 10 instead of e. Thus, log 10 = log₁₀ 10 = 1 .
The characteristic part of a logarithm may always be written down by inspection. Table is not required. Only for mantissa, we need a table. For most practical purposes, a system which gives the logarithms to four places of decimals, the Four-figure Tables, furnish sufficiently accurate results.
The given number is 0.0054 and it is a fraction, that is it is less than 1. Therefore, its characteristic is negative and its value is numerically greater than one than the number of ciphers (zeroes) immediately after the decimal point. As the number of zeros immediately after the decimal point is 2, characteristic of 0.0054 = -3 .
A simple way to understand this is to write the number in powers of 10 as follows.
log (0.0054) = log (5.4 x 10ˉ³) = log 5.4 + log 10ˉ³ = log 10ˉ³ + log 5.4 = -3 + .7324
Note that I have written the number in such a way that the decimal point stands after the first significant digit, namely, 5. When a number is written in this form , the characteristic is given at once by the index of the power of 10. Here it is -3 . Let us write some more numbers with identical number of significant digits as in the given number and find out their characteristics.
.00054, .054, .54, 5.4, 54, 540, 5400, 54000
Taking the logarithms of the above numbers,
log .00054 = log (5.4 x 10ˉ⁴) = log 5.4 + log 10ˉ⁴ =log 10ˉ⁴ + log 5.4 = -4 + .7324
log (.054) = log (5.4 x 10ˉ²) = log 5.4 + log 10ˉ² = log 10ˉ² + log 5.4 = -2 + .7324
log (.54) = log (5.4 x 10ˉ¹) = log 5.4 + log 10ˉ¹ = log 10ˉ¹ + log 5.4 = -1 + .7324
log (5.4)=log (5.4 x 10°)=log 5.4+log 10° =log 10°+log 5.4=0.log 10+.7324=0+.7324
log (54)=log (5.4 x 10¹)=log 5.4+log 10¹ =log 10¹+log 5.4=1.log 10+.7324=1+.7324
log (540)=log (5.4 x 10²)=log 5.4+log 10² =log 10²+log 5.4=2.log 10+.7324=2+.7324
log (5400)=log (5.4 x 10³)=log 5.4+log 10³ =log 10³+log 5.4=3.log 10+.7324=3+.7324
log (54000)=log (5.4 x 10⁴)=log 5.4+log 10⁴=log 10⁴+log 5.4=4.log10+.7324=4+.7324
From this example, we infer that the logarithms of all numbers which have the same sequence of digits, but differing only in the position of the decimal point, can be written so that they have all the same positive mantissa, but their characteristics are different, and they may be positive, negative, or zero as in the example above.
Answer:
Concept :
The logarithm is exponentiation's opposite function in mathematics. This indicates that the exponent to which a set number, base b, must be increased in order to create a specific number x, is represented by the logarithm of that number. The logarithm, in its most basic form, counts the number of times the same factor appears when multiplied repeatedly; for instance, since 1000 = 10 x 10 x 10 = 103, its "logarithm base 10" is 3, or log10 (1000) = 3. When there is no possibility of mistake or when the base is irrelevant, as in big O notation, the logarithm of x to base b is written as logb (x), logb x, or even without the explicit base, log x.
Explanation:
Given:
Characteristics of 0.00023
= Characteristics of 2.3×10^-4
=log(2.3×10^-4)
=-3.6
=-4
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