Logarithm is defined as the power to which a number is raised to yield some other values. Logarithms are the inverse of exponents. There is a unique way of reading the logarithm expression. For example, bx = n is called as 'x is the logarithm of n to the base b.
There are two parts of the logarithm: Characteristic and Mantissa. The integral part of a logarithm is called 'Characteristic' and the decimal part which is non-negative is called 'Mantissa'. The characteristic can be negative but mantissa can't. For example log10(120) = 2.078 ( 2 is characteristic and .078 is mantissa).
Formulas of Logarithm
Logarithmic Expressions follow different properties. The different properties of logarithms are mentioned below:
Product Formula of logarithm is stated below,
- loga(mn) = logam + logan (Product property)
Quotient Formula of logarithm is stated below,
- loga(m/n) = logam - logan (Quotient property)
Power Formula of logarithm is stated below,
- loga(mn) = nlogam (Power property)
Base of the a Logarithm is changed using the formula,
- logba = (logca)/(logcb) (Change of Base Property)
Read More about Change of Base Formula.
Various others Logarithm Formulas are,
- logb(a√n) = 1/a logbn
- log of 1 = loga1 = 0
- logaa = 1 (Identity rule)
- logba= logbc => a= c (Equality rule)
- a^{log_ax}
= x (Raised to log)
Natural log
The natural logarithm of a number is its logarithm to the base 'e'. 'e' is the transcendental and irrational number whose value is approximately equal to 2.71828182. It is written as ln x. ln x = logex. It is a special type of logarithm, used for solving time and growth problems. It is also used for solving the equation in which the unknown appears as the exponent of some other quantity.
Note : The natural log is also a logarithm function hence it also follows all the logarithm formulas discussed above.
Log formulas are very useful for solving various mathematical problems and these formula are easily derived using laws of exponents. Now lets learn about the derivation of some log formulas in detail.
Product formula of log states that : logb (xy) = logb x + logb y
Let take, logb x = m and logb y = n...(i)
Now using definition of logarithm,
x = bm and y = bn
⇒ x.y = bm × bn = b(m + n) → {by a law of exponents, pm × pn = p(m + n)}
⇒ x.y = b(m + n)
Converting into logarithm form,
m + n = logb xy
from eq. (1) we get
logb (xy) = logb x + logb y
Quotient formula of log states that : logb (x/y) = logb x - logb y
Let take, logb x = m and logb y = n...(i)
Now using definition of logarithm,
x = bm and y = bn
⇒ x/y = bm / bn = b(m - n) → {by a law of exponents, am / an = a(m - n)}
Converting into logarithm form
m - n = logb (x/y)
from eq. (1) we get
logb (x/y) = logb x - logb y
Power formula of log state that : logb ax = x logb a
Let logb a = m....(i)
Now using definition of logarithm,
ax = (bm)x
⇒ ax = (bmx) {by a law of exponents, (am)n = amn}
Converting into logarithm form,
logb ax = m x
using eq. (i) we get
logb ax = x logb a
Change of base formula of log states that: logb a = (logc a) / (logc b)
Let, logb a = x, logc a = y, and logc b = z
In exponential forms,
a = bx ... (1)
a = cy ... (2)
b = cz ... (3)
From (1) and (2),
bx = cy
(cz)x = cy (from (3))
⇒ czx = cy
⇒ zx = y
⇒ x = y / z
Substituting values of x, y, and z back,
logb a = (logc a) / (logc b)
Applications of Logarithm
Various applications of Logarithm are,
- Logarithm is used for expressing larger values in a simple format.
- Logarithms are used to simplify complex mathematical calculations especially those which involve exponential values.
- Logarithm is used for measuring earthquake intensity.
- Logarithms are used in chemistry for measuring pH value.
- Logarithms are also used for modeling business applications
- Logarithm is used by scientists to determine the rate of radioactive decay
- Logarithm is used by economists for plotting the graphs.
Read More
Example 1: Solve log2(x) = 4
Solution:
log2(x) = 4
24 = x
x = 16
Example 2: Solve log2(8) = x
Solution:
log2(8) = x
⇒ 2x = 8
⇒ 2x = 23
⇒ x = 3
Example 3: Find the value of x if log6(x - 3) = 1.
Solution:
log6(x - 3) = 1
⇒ 61 = (x - 3)
⇒ x - 3 = 6
⇒ x = 9
Example 4: Find x if log(x - 2) + log(x + 2) = log21
Solution:
log(x - 2) + log(x + 2) = log21
⇒ log(x - 2) + log(x + 2) = 0 [log(1) =0]
⇒ log[(x - 2)(x + 2)] = 0 [Product Rule]
⇒ (x - 2)(x + 2) = 1 [Antilog(0) = 1]
⇒ x2 - 4 = 1
⇒ x2 = 5
⇒ x = ±√5 [Log of Negative Number is Not Defined]
⇒ x = √5
Example 5: Find the value of log9(59049).
Solution:
Given log9 (59049) [95= 59049]
= log9(9)5
= 5.log9(9) (identity rule i.e logaa]
= 5
Example 6: Express log10(5) + 1 in form of log10x
Solution:
Given log10(5) + 1
= log10(5) + log1010 [Identity Rule]
= log10(5 × 10) [Product Rule]
= log1050
Example 7: Find the value of x if log10(x2 - 15) = 1.
Solution:
log10(x2 - 15) = 1
log10(x2 - 15) = log1010 [Identity Rule]
Applying Antilog,
⇒ (x2 - 15) = 10
⇒ x2 = 25
⇒ x = ±5
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