Change Base Formula - Algebra-2

1. Fundamental Concepts

Definition: The change of base formula for logarithms allows us to express a logarithm in terms of logarithms with any other base.
Formula: For any positive numbers $a$ , $b$ , and $x$ (where $a \neq 1$ and $b \neq 1$ ), the change of base formula is given by: $$\log_a{x} = \frac{\log_b{x}}{\log_b{a}}$$
Common Bases: Often, we use base 10 or base $e$ (natural logarithm) for convenience.

2. Key Concepts

Change of Base Formula: $$\log_a{x} = \frac{\log_b{x}}{\log_b{a}}$$

Base Conversion: This formula allows converting between different bases, such as from base 2 to base 10.

Application: Used in various fields including computer science, engineering, and physics for solving complex logarithmic equations.

3. Examples

Easy : Direct Conversion to Calculate Values

Calculate $\log_4 8$ using the change base formula.

Solution: Choose natural logarithm (or common logarithm) as the new base:

$\log_4 8 = \frac{\ln 8}{\ln 4} = \frac{\ln 2^3}{\ln 2^2} = \frac{3\ln 2}{2\ln 2} = \frac{3}{2}$ (the $\ln 2$ terms cancel out, no need for calculator).

Medium : Conversion + Simplification with Logarithm Properties

Simplify $\log_2 5 \cdot \log_5 4$ using the change base formula.

Solution: Convert both logarithms to the same common base (e.g., natural logarithm):

$\log_2 5 = \frac{\ln 5}{\ln 2}$, $\log_5 4 = \frac{\ln 4}{\ln 5}$

Multiply the two expressions:

$\log_2 5 \cdot \log_5 4 = \frac{\ln 5}{\ln 2} \cdot \frac{\ln 4}{\ln 5} = \frac{\ln 4}{\ln 2} = \frac{\ln 2^2}{\ln 2} = \frac{2\ln 2}{\ln 2} = 2$

Hard : Conversion + Solving for Unknowns

If $\log_3 x = 2\log_9 4$, find the value of $x$.

Solution: First, unify the bases using the change base formula (convert $\log_9 4$ to base 3):

$\log_9 4 = \frac{\log_3 4}{\log_3 9} = \frac{\log_3 4}{2}$ (since $\log_3 9 = \log_3 3^2 = 2$)

Substitute back into the original equation:

$\log_3 x = 2 \cdot \frac{\log_3 4}{2} = \log_3 4$

Since logarithms with the same base are equal if their arguments are equal, $x = 4$.

4. Problem-Solving Techniques

Calculator Usage: Use a scientific calculator to find logarithms of non-standard bases.
Estimation Check: Before calculating, estimate the value to check if the final answer makes sense.
Practice with Different Bases: Practice converting between different bases to become proficient with the change of base formula.