Algebra & Trigonometry II Essentials
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Algebra & Trigonometry II Essentials - Editors of REA
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CHAPTER 12
LOGARITHMS
12.1 LOGARITHMS
, b ≠ 0, we can rewrite it in its logarithmic form so that x is in terms of its exponent y, y = logbx. This expression is read as y is equal to the logarithm to the base b of x.
If x, y and a are positive real numbers, a ≠ 1, and r is any real number, then
loga(xy) = logax + logay,
,
logaxr = r logax.
Common logarithms are logarithms with a base of 10. We omit the base when working with base 10. That is
logx = log10x.
The following formula will enable us to calculate non-common logarithms by using common logarithms.
For example, to find the value of log73, we use the above rule to obtain:
These values can be found by referring to log tables.
The antilogarithm is the number corresponding to a given logarithm. The cologarithm of a positive number is the logarithm of its reciprocal.
The common logarithm of any number is expressible as a combination of two parts:
the characteristic, which is the integral part;
the mantissa, which is the decimal part of the number.
To find the common logarithm of a positive number:
Express the number in scientific notation.
Determine the index of the number, which is the characteristic.
To find the mantissa, see a table of common logarithms of numbers.
E.g. Find the logarithm of 30,700.
Solution: First express 30,700 in scientific notation. 30,700 = 3.07 × 10⁴. 4 is the characteristic. To find the mantissa, see a table of common logarithms of numbers. The mantissa is 4871. Thus log 30,700 = 4 + .4871 = 4.4871.
To find the antilogarithm:
Use the logarithm table to find the number that corresponds to that specific mantissa.
Rewrite that number in standard form.
Use the characteristic as the index for the number in standard form.
E.g. Find Antilog10 0.8762 − 2.
Solution: Let N = Antilog10 0.8762 − 2. The following relationship between log and antilog exists; log10x=a is the equivalent of x = antilog10 a. Therefore,
log10N = 0.8762 − 2.
The characteristic is −2. The mantissa is 0.8762. The number that corresponds to this mantissa is 7.52. This number is found from a table of common logarithms, base 10. Therefore,
Therefore, N = Antilog100.8762 −2 = 0.0752.
12.2 LOGARITHMIC, EXPONENTIAL AND POWER FUNCTIONS
The function f(x) = xn is called a power function in x. An exponential function in x is of the form f(x) = ax. A logarithmic function in x is of the form .f(x) = logax.
An equation involving one or more unknowns in an exponent is called an exponential equation. 2x + 6x + 8y+7 = y is an exponential equation in two unknowns.
The following are some examples of the graphs of power, exponential and logarithmic functions.
Fig. 12.1 Exponential function
Fig. 12.2 Exponential function
Fig. 12.3
Fig. 12.4 Exponential, Power, and Logarithmic Functions
Fig. 12.5 Logarithmic function
CHAPTER 13
SEQUENCES AND SERIES
13.1 SEQUENCES
A set of numbers u1, u2, u3,. . . in a definite order of arrangement and formed according to a definite rule is called a sequence. Each number in the sequence is called a term of the sequence. If the number of terms is finite, it is called a finite sequence, otherwise it is called an infinite sequence.
A general term n can be obtained by applying a general law of formation, by which any term in the sequence can be obtained.
An arithmetic progression (A. P.) is a sequence of numbers where each term excluding the first is obtained from the preceding one by adding a fixed quantity to it. This constant amount is called the common difference.
Let
a = first term of progression
1 = last term
d = common difference
k = number of terms
Sn