Properties of Logarithms - Algebra-2

1. Fundamental Concepts

Definition: Logarithms are the inverse operations to exponentiation, just as subtraction is the inverse of addition and division is the inverse of multiplication.
Base: The base of a logarithm is the number that is raised to a power in the exponential form.
Common Logarithm: A logarithm with base 10, often denoted as $$\log_{10}(x)$$ or simply $$\log(x)$$ .
Natural Logarithm: A logarithm with base e (approximately 2.71828), often denoted as $$\ln(x)$$ .

2. Key Concepts

Product Rule: $$\log_b(xy) = \log_b(x) + \log_b(y)$$

Quotient Rule: $$\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$$

Power Rule: $$\log_b(x^p) = p \cdot \log_b(x)$$

3. Examples

Easy

Expand: $\log_2(5z)$

Solution: According to the "Product to Sum" property, $\log_2(5z)=\log_2 5 + \log_2 z$

Condense: $\log_4 32 - \log_4 2$

Solution: According to the "Quotient to Difference" property, $\log_4\frac{32}{2}=\log_4 16$ . Since $4^2=16$ , the result is 2.

Simplify: $\log_2(8)+log_2(4)$
Step-by-Step Solution:

Apply the product rule: $$\log_2(8) + \log_2(4) = \log_2(8 \cdot 4)$$
Simplify the argument: $$\log_2(32)$$
Since $$2^5 = 32$$ , we have: $$\log_2(32) = 5$$

Medium

Expand: $\ln\frac{x^3}{y^2}$

Solution: First, apply the "Quotient to Difference" property to get $\ln x^3 - \ln y^2$ , then apply the "Power to Multiple" property to get $3\ln x - 2\ln y$ .

Condense: $6\log_2 x + 5\log_2 y$

Solution: First, reverse the "Power to Multiple" property to convert the expression to $\log_2 x^6 + \log_2 y^5$ , then apply the "Product to Sum" property to get $\log_2(x^6y^5)$ .

Problem: Solve for x in $$\log_3(x^2) = 4$$

Step-by-Step Solution:

Apply the power rule: $$2 \cdot \log_3(x) = 4$$
Solve for $$\log_3(x)$$ : $$\log_3(x) = 2$$
Convert to exponential form: $$3^2 = x$$
Thus, $$x = 9$$

Hard

Expand: $\log_3\frac{2x^4}{5y}$

Solution:
Apply the "Quotient to Difference" property to get $\log_3(2x^4) - \log_3(5y)$ .
Then, apply the "Product to Sum" property to each term separately, and finally use the "Power to Multiple" property: $\log_3 2 + 4\log_3 x - \log_3 5 - \log_3 y$ .

Condense: $2\ln x - \ln 3 - 3\ln y + \ln 6$

Solution:
First, rearrange the "terms with coefficients" into $\ln x^2 - \ln y^3$ and the "constant terms" into $\ln 6 - \ln 3$ .
Then, apply the "Quotient to Difference" property to each part to get $\ln\frac{x^2}{y^3} + \ln 2$ .
Finally, apply the "Product to Sum" property to get $\ln\left(\frac{2x^2}{y^3}\right)$ .

4. Problem-Solving Techniques

Identify the Base: Always start by identifying the base of the logarithm to ensure consistency in your calculations.
Use Logarithmic Identities: Familiarize yourself with the properties of logarithms to simplify expressions and solve equations efficiently.
Check Your Work: After solving, substitute your solution back into the original equation to verify its correctness.