Page 2: Solving Exponential and Logarithmic Equations and Inequalities
This page delves into practical applications of exponential and logarithmic properties, focusing on solving equations and inequalities.
Example: To solve the equation 3ˣ = 81, we apply logarithms to both sides:
log(3ˣ) = log(81)
x × log(3) = log(81)
x = log(81) / log(3)
This example demonstrates the use of propriété ln et exp to solve exponential equations by converting them to linear equations using logarithms.
For inequalities, the process is similar but requires careful attention to the direction of the inequality when applying logarithms.
Highlight: When dealing with logarithmic inequalities, it's crucial to consider the sign of the logarithm's argument, as it can change the direction of the inequality.
The page provides an example of solving the inequality 9ˣ < 4ˣ × 0.32ˣ:
- Apply logarithms to both sides: log(9ˣ) < log(4ˣ × 0.32ˣ)
- Use logarithm properties to simplify: x × log(9) < x × log(4) + x × log(0.32)
- Factor out x: x × (log(9) - log(4) - log(0.32)) < 0
- Solve for x, considering the sign of the coefficient
Vocabulary: An equation results in a specific value or set of values, while an inequality results in an interval or range of values.
The page concludes by emphasizing the difference between equations and inequalities in terms of their solutions, reinforcing the importance of understanding these concepts for fonction exponentielle exercices corrigés PDF.
