Answer:
### Properties of Logarithms
- Product Rule:
[tex]\log_b(x \cdot y) = \log_b(x) + \log_b(y)[/tex]
- Quotient Rule:
[tex]\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)[/tex]
- Power Rule:
[tex]\log_b(x^y) = y \cdot \log_b(x)[/tex]
### Examples
For a common logarithm:
[tex]\[
\log_{10}(100) = 2 \quad \text{because} \quad 10^2 = 100
\][/tex]
For a natural logarithm:
[tex]\ln(e) = 1 \quad \text{because} \quad e^1 = e[/tex]
### Conversion between Logarithms
To convert a logarithm from one base to another, you can use the change of base formula:
[tex]log_b(x) = \frac{\log_k(x)}{\log_k(b)}[/tex]
where ( k ) is any positive number different from 1.
### Example of Change of Base Formula
[tex]\log_2(8) = \frac{\log_{10}(8)}{\log_{10}(2)}[/tex]
### Solving a Logarithmic Equation
[tex]( \log_b(x) = y \):[/tex]
1. Rewrite the equation in its exponential form,
[tex]\( b^y = x \).[/tex]
2. Solve for ( x ).
[tex]For \:example, if \( \log_5(x) = 3 \):[/tex]
[tex]5^3 = x \implies x = 125[/tex]
### Summary Table
Type Of Logarithm Base Notation
Common Logarithm 10 Log_10(x)
Natural Logarithm e log_e(x) or
in(x)
