91Ó°ÊÓ

To solve logarithmic equations, understanding the properties of logarithms is crucial. These properties help simplify and manipulate the equations. The three primary properties include:

• **Product Property**: \( \log_b (MN) = \log_b M + \log_b N \)
• **Quotient Property**: \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \)
• **Power Property**: \( \log_b (M^k) = k \cdot \log_b M \)

These properties are essential tools. They simplify complex logarithmic expressions and make solving equations more manageable. Always consider these properties as a starting point.

Solving logarithmic equations involves isolating the logarithm or exponent. In the given exercise:

\( 2 \cdot \log (x) = \log (20 - x) \)

Use the Power Property of logarithms to simplify the left side:
\( \log (x^2) = \log (20 - x) \)

Since the logs are equal, you can set their arguments equal to each other:
\( x^2 = 20 - x \)

This transforms the logarithmic equation into a quadratic equation, ready for further manipulation. Ensure you check for any extraneous solutions by substituting them back into the original equation.

Once you have a transformed equation, the next step is solving it through algebraic manipulation.

For example, from the previous section, you have:
\( x^2 = 20 - x \)

Rearrange to form a standard quadratic equation:
\( x^2 + x - 20 = 0 \)

You can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Here, it factors to:
\( (x + 5)(x - 4) = 0 \)

So, \( x = -5 \) or \( x = 4 \). However, log functions are only defined for positive values. Substituting back, you'll find:

• \( x = -5 \) is invalid
• \( x = 4 \) is valid

Thus, the solution to the equation \( 2 \cdot \log (x) = \log (20 - x) \) is \( x = 4 \). Understanding and applying these manipulation techniques is essential for solving logarithmic equations effectively.