Formulas
Logarithm:
If $a^{x}$ = N, then, $log_{a}N$ = x
Important formulae in logarithm:
i.$log_{a}a$ = 1
ii.$log_{a}1$ = 0
iii.$log_{a}(mxn)$ = $log_{a}m + log_{a}n$
iv.$log_{a}(m/n)$ = $log_{a}m - log_{a}n$
v.$log_{a}m^p$ = $p \times log_{a}m$
vi.$log_{a}m$ = $log_{b}m / log_{a}b$
vii.$a ^{log b}$ = $b^{log a}$
Logarithm:
The logarithm of any number to a given base is the power to which the base must be raised in order to equal the given number.
If $a^{x}$ = N, then, $log_{a}N$ = x
Example:
64
By using this formla,If $a^{x}$ = N, then, $log_{a}N$ = x
64 = $4^{3}$ can be expressed as $log_{4}64$ = 3.
Exercise:
Exponential form: $b^{y}$ = x
Logarithm form: $log_{b} (x)$ = y
Given: $log_{2} 64$
Let the solution of $log_{2} 64$ be y.
$Log_{2} 64$ = y
$2^{y}$ = 64
$2^{6}$ = 64
Therefore, y = 6
Common logarithm:
Logarithms with base 10 are common logarithms.
Common logarithms are written as $log_{10}x$ and if any expression is not indicated with a base, then base 10 is considered.
On a calculator it is the "log" button.
Example:
Evaluate log(1000)
if any expression is not indicated with a base, then base 10 is considered.
log(1000)
= $log_{10}(1000)$
= 3
Exercise:
$log_{10}(100)$ ... ?
$10^{2}$ = 100
So an exponent of 2 is needed to make 10 into 100, and:
$log_{10}(100)$ = 2
Natural logarithm:
Logarithms with base e are natural logarithms.
e is an irrational number, approximately 2.71828183
Natural logarithms are written as $log_{e}x$ and denoted as ln (x).
On a calculator it is the "ln" button.
Example:
ln(7.389)
ln(7.389)= $log_{e}(7.389)$ ≈ 2
Because $2.71828^2$ ≈ 7.389
Exercise:
Logarithm of a number contains 2 parts: Characteristic and Mantissa
Characteristic
Characteristic is an integral part of logarithm.
Case 1:
If number is greater than 1.
In this condition, characteristic is considered as one less than the number of digits in the left of decimal point in the given number.
Example:
Calculate characteristic of the logarithm 256.23
Number of digits to the left of decimal point are 3. Hence, the characteristic is one less than number of digits before decimal points i.e 2.
Exercise:
In this condition, characteristic is considered as one less than the number of digits in the left of decimal point in the given number.
so,Characteristic = 3 -1 = 2
Case 2:
If number is less than 1.
In this condition, characteristic is considered as one more than the number of zeros between decimal point and first digit of the number. It is negative and is denoted as (One bar) $\overline{1}$ or (Two bar) $\overline{2}$
Example:
Calculate characteristic of logarithm 0.00735
Number of zeros between decimal point and first significant digit 7 are 2. Hence the characteristic is one more than number of zeros i.e 3
Exercise:
In this condition, characteristic is considered as one more than the number of zeros between decimal point and first digit of the number.
So,Characteristic = 2 +1 = 3
Mantissa:
It is the decimal part of logarithm. Log table is used to find the mantissa.
Example:
i.log 2 = 0.3010, here the mantissa is 0.3010
ii.log 200 = 2.3010, here the mantissa is 0.3010
Exercise:
Mantissa is the decimal part of logarithm.
so ,here the mantissa is 0.3010
Important formulae in logarithms:
i.$log_{a}a$ = 1
ii.$log_{a}1$ = 0
iii.$log_{a}(mxn)$ = $log_{a}m + log_{a}n$
iv.$log_{a}(m/n)$ = $log_{a}m - log_{a}n$
v.$log_{a}m^p$ = $p \times log_{a}m$
vi.$log_{a}m$ = $log_{b}m / log_{a}b$
vii.$a ^{log b}$ = $b^{log a}$
Example:
$Log_{2}8$ = $log_{2}(4 × 2)$
By using this formula,$Log_{a}(MN)$ = $log_{a}M + log_{a}N$
=> $log_{2}4 + log_{2}2$
=> 2 + 1 = 3
Example:
$Log_{2}8$ = $log_{2}(16 ÷ 2)$
By using this formula,$Log_{a}(M/N)$ = $log_{a}M - log_{a}N$
=> $log_{2}16 - log_{2}2$
=> 4 - 1 = 3
Exercise:
Given expression = log 8 + log (1/8)
= log 8 x (1/8)
= log 1
= 0