Logarithm Formula
Logarithm was invented in the 17th century by Scottish mathematician John Napier (1550-1617). The Napier logarithm was the first to be published in 1614. Henry Briggs introduced a common (base 10) logarithm. John Napierβs purpose was to assist in the multiplication of quantities that were called sines.
Table of Content
- Logarithm Formula
- Properties of Logarithm
- Product Formula of Logarithms
- Quotient Formula of Logarithms
- Power Formula of Logarithms
- Change of Base Formula
- Other Logarithm Formulas
- Properties of Natural Log
- Product Rule
- Quotient Rule
- Reciprocal Rule
- Log of Power
- Natural Log of e
- Log of 1
- Log Formulas Derivation
Logarithm Formula
A 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 βx 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 βCharacteristicβ and the decimal part which is non-negative is called βMantissaβ. The characteristic can be negative but mantissa canβt. For example log10(120) = 2.078 ( 2 is characteristic and .078 is mantissa).
Properties of Logarithm
Logarithmic Expressions follow different properties. The different properties of logarithms are mentioned below:
Product Formula of Logarithms
Product Formula of logarithm is stated below,
- loga(mn) = logam + logan (Product property)
Quotient Formula of Logarithms
Quotient Formula of logarithm is stated below,
- loga(m/n) = logam β logan (Quotient property)
Power Formula of Logarithms
Power Formula of logarithm is stated below,
- loga(mn) = nlogam (Power property)
Change of Base Formula
Base of the a Lograthin is changed using the formula,
- logba = (logca)/(logcb) (Change of Base Property)
Read More about Change of Base Formula.
Other Logarithm Formulas
Various others Logarithm Formulas are,
- logb(aβn) = 1/a logbn
- log of 1 = loga1 = 0
- logaa = 1 (Identity rule)
- logba= logbc => a= c (Equality rule)
- [Tex]a^{log_ax} [/Tex] = x (Raised to log)
Natural log
The natural logarithm of a number is its logarithm to the base βeβ. βeβ 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.
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Properties of Natural Log
Properties of Natural Log are,
Product Rule
The product rule of natural log states that,
ln(xy) = ln(x) + ln(y)
Quotient Rule
The quotient rule of natural log states that,
ln(x/y) = ln(x) β ln(y)
Reciprocal Rule
The reciprocal rule of natural log states that,
ln(1/x) = -ln(x)
Log of Power
The log of any term that is written in power term is written as,
ln(xy) = y ln(x)
Natural Log of e
The natural log of βeβ is always 1(one) as the base in natural log is βeβ. This is represented as,
ln (e) = 1
Log of 1
The log of 1 is always zero.
ln (1) = 0
Log Formulas Derivation
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.
Derivation of Product Formula of Log
Product formula of log states that,
logb (xy) = logb x + logb y
This is derived as,
Let take, logb x = m and logb y = nβ¦(i)
Now using definition of logarithm,
x = bm and y = bn
β x.y = bm Γ bn = b(m + n) β {by a law of exponents, pm Γ pn = p(m + n)}
β x.y = b(m + n)
Converting into logarithm form,
m + n = logb xy
from eq. (1)
logb (xy) = logb x + logb y
Derivation of Quotient Formula of Log
Quotient formula of log states that,
logb (x/y) = logb x β logb y
This is derived as,
Let take, logb x = m and logb y = nβ¦(i)
Now using definition of logarithm,
x = bm and y = bn
β x/y = bm / bn = b(m β n) β {by a law of exponents, am / an = a(m β n)}
Converting into logarithm form
m β n = logb (x/y)
from eq. (1)
logb (x/y) = logb x β logb y
Derivation of Power Formula of Log
Power formula of log state that,
logb ax = x logb a
This is derived as,
Let logb a = mβ¦.(i)
Now using definition of logarithm,
ax = (bm)x
β ax = (bmx) {by a law of exponents, (am)n = amn}
Converting into logarithm form,
logb ax = m x
using eq. (i)
logb ax = x logb a
Derivation of Change of Base Formula of Log
Change of base formula of log states that,
logb a = (logc a) / (logc b)
This is derived as,
Let, logb a = x, logc a = y, and logc b = z
In exponential forms,
a = bx β¦ (1)
a = cy β¦ (2)
b = cz β¦ (3)
From (1) and (2),
bx = cy
(cz)x = cy (from (3))
β czx = cy
β zx = y
β 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 value.
- Logarithm is Used for measuring earthquake intensity.
- Logarithm is Used for measuring pH value.
- Logarithm is 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.
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Solved Examples on Logarithm Formula
Example 1: Solve log2(x) = 4
Solution:
log2(x) = 4
24 = x
x = 16
Example 2: Solve log2(8) = x
Solution:
log2(8) = x
β 2x = 8
β 2x = 23
β x = 3
Example 3: Find the value of x if log6(x β 3) = 1.
Solution:
log6(x β 3) = 1
β 61 = (x β 3)
β x β 3 = 6
β x = 9
Example 4: Find x if log(x β 2) + log(x + 2) = log21
Solution:
log(x β 2) + log(x + 2) = log21
β log(x β 2) + log(x + 2) = 0 [log(1) =0]
β log[(x β 2)(x + 2)] = 0 [Product Rule]
β (x β 2)(x + 2) = 1 [Antilog(0) = 1]
β x2 β 4 = 1
β x2 = 5
β x = Β±β5 [Log of Negative Number is Not Defined]
β x = β5
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 Γ 10) [Product Rule]
= log1050
Example 7: Find the value of x if log10(x2 β 15) = 1.
Solution:
log10(x2 β 15) = 1
log10(x2 β 15) = log1010 [Identity Rule]
Applying Antilog,
β (x2 β 15) = 10
β x2 = 25
β x = Β±5
Practice Questions on Logarithm Formula
Q1. Find the value of x: 3.log(x) = log 27
Q2. Simplify log2(16) + 2.log3(9)
Q3. Find the value of x: 2.log(2x) = log 81
Q4. Simplify log3(9) β 3.log3(27)
Q5. Simplify ln(x3/y2z)
FAQs on Logarithm Formula
What are Logarithm Formulas?
Logarithm Formulas are the formulas that are useful to solve the logarithmic problems. They are derived using laws of exponents.
How To Derive Log Formulas?
Logarithm Formulas are derived using Laws of Exponetnts
What are Applications of Log Formulas?
Various applications of log formulas are,
- They are used to simplify log problems.
- They are used to solve various large calculation.
- They are used to find the Derivative and Integral of various functions.
- They are used in graph plotting, etc.
What is Product Formula of Logarithm?
The product formula of logarithm states that, for any base βnβ, logn(a.b) = logn(a) + logn(b)
What is Quotient Formula of Logarithm?
The quotient formula of logarithmic states that, for any base βnβ, logn(a/b) = logn(a) β logn(b)
What is Power Formula of Logarithm?
The power formula of logarithm states that, for any base βnβ, logn(a)b = b.logn(a)
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