“logarithm”的概念、定义、翻译、参考文献-科学参考

logarithm

Physics

The power to which a number, called the base, has to be raised to give another number. Any number y can be written in the form y=xⁿ. n is then the logarithm to the base x of y, i.e. n=log_xy. If the base is 10, the logarithms are called common logarithms. Natural (or Napierian) logarithms (named after John Napier; 1550–1617) are to the base e=2.718 28…, written log_ey or lny. Logarithms were formerly used to facilitate calculations, before the advent of electronic calculators.

A logarithm contains two parts, an integer and a decimal. The integer is called the characteristic, and the decimal is called the mantissa. For example, the logarithm to the base 10 of 210 is 2.3222, where 2 is the characteristic and 0.3222 is the mantissa.

Mathematics

Let a, x be positive numbers with a ≠ 1. Then log_ax, the logarithm of x to base a, is defined by

${log}_{a} x = \frac{ln x}{ln a},$

where ln denotes the logarithmic function. This is equivalent to saying that y = log_ax if and only if x = a^y or that the function log_ax is the inverse function of a^x. The following properties hold, where x, y, and r are real, with x,y > 0:
1. (i) log_a(xy) = log_a x + log_a y.
2. (ii) log_a(1/x) = −log_a x.
3. (iii) log_a(x^r) = r log_a x.
4. (iv) Logarithms to different bases are related by the formula
  
  ${log}_{b} x = \frac{{log}_{a} x}{{log}_{a} b} .$
5. (v) $\frac{d}{d x} {log}_{a} x = \frac{1}{x ln a} .$
Logarithms to base 10 are called common logarithms. Logarithms to base e are called natural logarithms. See complex logarithm.

Statistics

The logarithm to base a (>0) of a positive number b is denoted by log_a b=x, where a^x=b. Logarithms have the fundamental property log_a (bc)=log_a b+log_a c. A logarithm to base 10 is usually denoted by log and a natural logarithm, i.e. a logarithm to base e (see exponential), by ln. Before the advent of computers and calculators, logarithms were used in evaluating products.

Chemical Engineering

Any real number can be written as another number raised to a power in the form $y = x^{n}$ where n is the the logarithm to the base x of y, i.e. $n = \log_{x} y$ . If base 10 is used, the logarithms are called common logarithms. Natural logarithms or (Naperian logarithms) are written to the base e = 2.718 28. . .and written as either log_e or ln and named after Scottish mathematician John Napier (1550–1617). Logarithms contain two parts: the characteristic is the integer and the mantissa is the decimal. For example, the logarithm to the base e of 10 is 2.302 where 2 is the characteristic and 0.302 is the mantissa. Note that for any base, the logarithm of 1 is zero, the logarithm of 0 is not defined, the logarithm of a number greater than 1 is positive, the logarithm of a number between 0 and 1 is negative, and the logarithm of a negative number cannot be evaluated as a real number. In the past, tables were constructed called Tables of Logarithms. Nowadays, electronic calculators have superseded the use of these tables.

单词	logarithm
释义	logarithm Physics The power to which a number, called the base, has to be raised to give another number. Any number y can be written in the form y=xⁿ. n is then the logarithm to the base x of y, i.e. n=log_xy. If the base is 10, the logarithms are called common logarithms. Natural (or Napierian) logarithms (named after John Napier; 1550–1617) are to the base e=2.718 28…, written log_ey or lny. Logarithms were formerly used to facilitate calculations, before the advent of electronic calculators. A logarithm contains two parts, an integer and a decimal. The integer is called the characteristic, and the decimal is called the mantissa. For example, the logarithm to the base 10 of 210 is 2.3222, where 2 is the characteristic and 0.3222 is the mantissa. Mathematics Let a, x be positive numbers with a ≠ 1. Then log_ax, the logarithm of x to base a, is defined by ${log}_{a} x = \frac{ln x}{ln a},$ where ln denotes the logarithmic function. This is equivalent to saying that y = log_ax if and only if x = a^y or that the function log_ax is the inverse function of a^x. The following properties hold, where x, y, and r are real, with x,y > 0: (i) log_a(xy) = log_a x + log_a y. (ii) log_a(1/x) = −log_a x. (iii) log_a(x^r) = r log_a x. (iv) Logarithms to different bases are related by the formula ${log}_{b} x = \frac{{log}_{a} x}{{log}_{a} b} .$ (v) $\frac{d}{d x} {log}_{a} x = \frac{1}{x ln a} .$ Logarithms to base 10 are called common logarithms. Logarithms to base e are called natural logarithms. See complex logarithm. Statistics The logarithm to base a (>0) of a positive number b is denoted by log_a b=x, where a^x=b. Logarithms have the fundamental property log_a (bc)=log_a b+log_a c. A logarithm to base 10 is usually denoted by log and a natural logarithm, i.e. a logarithm to base e (see exponential), by ln. Before the advent of computers and calculators, logarithms were used in evaluating products. Chemical Engineering Any real number can be written as another number raised to a power in the form $y = x^{n}$ where n is the the logarithm to the base x of y, i.e. $n = \log_{x} y$ . If base 10 is used, the logarithms are called common logarithms. Natural logarithms or (Naperian logarithms) are written to the base e = 2.718 28. . .and written as either log_e or ln and named after Scottish mathematician John Napier (1550–1617). Logarithms contain two parts: the characteristic is the integer and the mantissa is the decimal. For example, the logarithm to the base e of 10 is 2.302 where 2 is the characteristic and 0.302 is the mantissa. Note that for any base, the logarithm of 1 is zero, the logarithm of 0 is not defined, the logarithm of a number greater than 1 is positive, the logarithm of a number between 0 and 1 is negative, and the logarithm of a negative number cannot be evaluated as a real number. In the past, tables were constructed called Tables of Logarithms. Nowadays, electronic calculators have superseded the use of these tables.
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