1. Logarithms: If [latex] a [/latex] is a positive real number, other than 1 and [latex] a^m [/latex] = [latex] x [/latex], then we write: [latex] m [/latex] = [latex] log_{a}x [/latex] and say that the value of \( log x \) to the base \( a \) is \( m \).
Example: \( (10)^3 \)= 1000 ⇒ [latex] log_{10}1000 [/latex] = 3
2.Common logarithms: Logarithms to the base 10 are known as common logarithms.
3.Characteristic: The integral part of the logarithm of a number is called its characteristic. For example: [latex] log_{10}15 [/latex] = 1.176 = 1+0.176. So the integer part is 1 and so the characteristic is 1. The characteristic of the logarithm of a number is an integer, negative or positive. For example: [latex] log_{10}0.5 [/latex] = -0.301 = -1+0.699. Here the characteristic is 1.
(i) When the number is greater than 1. Here, the characteristic is one less than the number of digits in the left of the decimal point in the given number.
Examples: [latex] log_{10}15 [/latex] = 1.176 = 1+0.176. Number of digits in 15.0 to the left of decimal is 2. So characteristic is 2-1=1.
[latex] log_{10}183.5 [/latex] = 2.2636 = 2+0.2636.  Number of digits in 183.5 to the left of decimal is 3. So characteristic is 3-1=2.
(ii) When the number is positive and less than 1. Here, the characteristic is one more than the number of zeros between the decimal point and the first significant digit of the number and it is negative. The negative number can also be written as: instead of -1, it is written as [latex]\bar{1}[/latex].
Examples: [latex] log_{10}0.6 [/latex] = -0.2218= -1+0.7782. Number of zeroes in 0.5 to the right of decimal is 1. So characteristic is 0+1=1 but with a negative sign. i.e. characteristic is -1.
[latex] log_{10}0.008[/latex] = -2.09691001301 = -3+0.9031.  Number of zeroes in 0.008 to the left of decimal is 2. So characteristic is 2+1=3, but with a negative sign. i.e. characteristic is -3.
4.Mantissa: Decimal part of the algorithm of a number is known as its mantissa. Mantissa can never be negative. For example: [latex] log_{10}15 [/latex] = 1.176 = 1+0.176. The decimal part is 0.176. So, the Mantissa is 0.176. There is also a log table to look for the mantissa.

1. Product: [latex]log_{b}(a . c)[/latex] = [latex]log_{b}a[/latex] + [latex]log_{b}c[/latex]


2. Quotient: [latex]log_{b}\frac{a}{b}[/latex] = [latex]log_{b}a[/latex] - [latex]log_{b}c[/latex]


3. Power: [latex]log_{b}a^{n}[/latex] = [latex]n . log_{b}a[/latex]


4. [latex]log_{b}1 = 0[/latex]


5. [latex]log_{b}b = 1[/latex]


6. Inverse 1: [latex]log_{b}b^{n}[/latex] = [latex]n[/latex]


7. Inverse 2: [latex]b{log_{b}n}[/latex] = [latex]n, n > 0[/latex]


8. One-to-One: [latex]log_{b}a[/latex] = [latex]log_{b}c[/latex] if and only if a = c


9. Change of Base: [latex]log_{b}a[/latex] = [latex]\frac{log_{c}a}{log_{c}b}[/latex] = [latex]\frac{log a}{log b}[/latex] = [latex]\frac{In a}{In b}[/latex]

Example 1 Use the properties of logarithms to rewrite expression as a single logarithm:
[latex]2log_{b}x[/latex] + [latex]\frac{1}{2}log_{b}(x + 4)[/latex]
Solution:

[latex]2log_{b}x[/latex] + [latex]\frac{1}{2}log_{b}(x + 4)[/latex]
[latex]log_{b}x^{2}[/latex] + [latex]log_{b}(x + 4)^{\frac{1}{2}}[/latex] [Using Power Property]
[latex]log_{b}[x^{2}(x + 4)^{\frac{1}{2}}][/latex] [Using Product Property]

Example 2 Use the properties of logarithms to rewrite expression as a single logarithm:
[latex]4log_{b}(x + 2) - 3log_{b}(x - 5)[/latex]
Solution:

[latex]4log_{b}(x + 2) - 3log_{b}(x - 5)[/latex]
= [latex]log_{b}(x + 2)^{4} - log_{b}(x - 5)^{3}[/latex] [Using Power Property]
= [latex]log_{b}\frac{(x + 2)^{4}}{(x - 5)^{3}}[/latex] [Using Quotient Property]

Example 3 Use the properties of logarithms to express the following logarithm in terms of logarithm of [latex]x[/latex], [latex]y[/latex] and [latex]z[/latex].
[latex]log_{b}(xy^{2})[/latex]
Solution:

[latex]log_{b}(xy^{2})[/latex] = [latex]log_{b}x[/latex] + [latex]log_{b}y^{2}[/latex] [Using Product Property]
= [latex]log_{b}x[/latex] + [latex]2log_{b}y[/latex] [Using Power Property]

Example 4 Use the properties of logarithms to express the following logarithm in terms of logarithm of [latex]x[/latex], [latex]y[/latex] and [latex]z[/latex].
[latex]log_{b}\frac{x^{2}\sqrt{y}}{z^{5}}[/latex]
Solution:

[latex]log_{b}\frac{x^{2}\sqrt{y}}{z^{5}}[/latex]
= [latex]log_{b}(x^{2}\sqrt{y}) \ - log_{b}z^{5}[/latex] [Using Quotient Property]
= [latex]log_{b}x^{2}[/latex] + [latex]log_{b}\sqrt{y}[/latex] - [latex]log_{b}z^{5}[/latex] [Using Product Property]
= [latex]2log_{b}x[/latex] + [latex]\frac{1}{2}log_{b}y[/latex] - [latex]5log_{b}z[/latex] [Using Power Property]

Properties of logarithms:

1. Product: [latex]log_{b}(a . c)[/latex] = [latex]log_{b}a[/latex] + [latex]log_{b}c[/latex]


2. Quotient: [latex]log_{b}\frac{a}{b}[/latex] = [latex]log_{b}a[/latex] - [latex]log_{b}c[/latex]


3. Power: [latex]log_{b}a^{n}[/latex] = [latex]n . log_{b}a[/latex]


4. [latex]log_{b}1 = 0[/latex]


5. [latex]log_{b}b = 1[/latex]


6. Inverse 1: [latex]log_{b}b^{n}[/latex] = [latex]n[/latex]


7. Inverse 2: [latex]b{log_{b}n}[/latex] = [latex]n, n > 0[/latex]


8. One-to-One: [latex]log_{b}a[/latex] = [latex]log_{b}c[/latex] if and only if a = c


9. Change of Base: [latex]log_{b}a[/latex] = [latex]\frac{log_{c}a}{log_{c}b}[/latex] = [latex]\frac{log a}{log b}[/latex] = [latex]\frac{In a}{In b}[/latex]

1. Calculate the value of: [latex] log_{3}27 [/latex]?
Solution:

Given that,
[latex] log_{3}27 [/latex]
Let [latex] log_{3}27 [/latex] = n
Then, [latex]3^n[/latex] = 27
⇒ [latex]3^n[/latex] = [latex]3^3[/latex]
⇒ n = 3
Therefore, [latex] log_{3}27 [/latex] = 3

2. If [latex] log_{\sqrt{8}}x [/latex] = 3 [latex]\frac{1}{3}[/latex], find the value of [latex]x[/latex]?
Solution:

Given that,
[latex] log_{\sqrt{8}}x [/latex] = 3 [latex]\frac{1}{3}[/latex]
⇒ [latex] log_{\sqrt{8}}x [/latex] = [latex]\frac{10}{3}[/latex]
⇒ [latex] x [/latex] = [latex](\sqrt{8})^{\frac{10}{3}}[/latex]
⇒ [latex]2^{\frac{3}{2} * \frac{10}{3}}[/latex]
⇒ [latex]2^{5}[/latex]
⇒ 32
Therefore, the value of [latex]x[/latex] =32

3. Simplify: [latex]log \frac{75}{16}[/latex] - 2[latex]log \frac{5}{9}[/latex] + [latex]log \frac{32}{243}[/latex]?
Solution:

Given that,
[latex]log \frac{75}{16}[/latex] - 2[latex]log \frac{5}{9}[/latex] + [latex]log \frac{32}{243}[/latex]
⇒ [latex]log \frac{75}{16}[/latex] - [latex]log (\frac{5}{9})^2[/latex] + [latex]log \frac{32}{243}[/latex]
⇒ [latex]log \frac{75}{16}[/latex] - [latex]log (\frac{32}{243})[/latex] + [latex]log \frac{32}{243}[/latex]
⇒ log 2

4. If [latex] log_{10}2 [/latex] = 0.30103, find the value of [latex] log_{10}50 [/latex]?
Solution:

Given that,
[latex] log_{10}2 [/latex] = 0.30103
Then, [latex] log_{10}50 [/latex] = [latex] log_{10}\frac{100}{2} [/latex] = [latex] log_{10}100 [/latex] - [latex] log_{10}2 [/latex]= 2 - 0.30103 = 1.69897
Therefore, [latex] log_{10}50 [/latex] = 1.69897

5. Simplify : [latex]\frac{1}{log_{xy}xyz} + \frac{1}{log_{yz}xyz} + \frac{1}{log_{zx}xyz}[/latex]?
Solution:

Given that,
[latex]\frac{1}{log_{xy}xyz} + \frac{1}{log_{yz}xyz} + \frac{1}{log_{zx}xyz}[/latex]
⇒ [latex]log_{xyz}xy + log_{xyz}yz + log_{xyz}xz[/latex]
⇒ [latex]log_{xyz}(xy * yz * zx)[/latex]
⇒ [latex]log_{xyz}(xyz)^2[/latex]
⇒ 2 [latex]log_{xyz}(xyz)[/latex]
⇒ 2
Therefore, [latex]\frac{1}{log_{xy}xyz} + \frac{1}{log_{yz}xyz} + \frac{1}{log_{zx}xyz}[/latex] = 2

Definition of Logarithms
y = [latex]\log_{a}[/latex] x if and only if x = [latex]a^y[/latex], a [latex]>[/latex] 0, a [latex]\neq[/latex] 1. Properties of Logarithms:
1. [latex]\log_{a}[/latex] 1 = 0
2. [latex]\log_{a}[/latex] a = 1
3. [latex]\log_{a}[/latex] 0 = [latex]\begin{cases} - \infty\;\;if\;\;a > 1\\ + \infty\;\;if\;\;a < 1 \end{cases}[/latex]
4. [latex]\log_{a}[/latex] (xy) = [latex]\log_{a}[/latex] x + [latex]\log_{a}[/latex] y
5. [latex]\log_{a}\;\frac{x}{y}[/latex] = [latex]\log_{a}[/latex] x - [latex]\log_{a}[/latex] y
6. [latex]\log_{a}\;\sqrt[n]{x}[/latex] = [latex]\frac{1}{n}[/latex] [latex]\log_{a}[/latex] x
7. [latex]\log_{a}\;x[/latex] = [latex]\frac{\log_{c}\;x}{\log_{c}\;a}[/latex] = [latex]\log_{c}\;x[/latex] [latex]\cdot[/latex] [latex]\log_{a}\;c[/latex], c [latex]>[/latex] 0, c [latex]\neq[/latex] 1
8. [latex]\log_{a}\;c[/latex] = [latex]\frac{1}{\log_{c}\;a}[/latex]
9. x = [latex]a^{\log_{a}\;x}[/latex]
10. Logarithm to Base 10       [latex]\log_{10}\;x[/latex] = log x
11. Natural Logarithm       [latex]\log_{e}\;x[/latex] = ln x,       where e = [latex]\displaystyle{\lim_{x \to \infty}}(1 + \frac{1}{k})^k[/latex] = 2.718281828...
12. log x = [latex]\frac{1}{ln 10}[/latex]ln x = 0.434294 ln x
12. ln x = [latex]\frac{1}{log\;e}[/latex]log x = 2.302585 log x