SUMMARY
The equation 6 \cdot e2x = 4x can be solved by taking the natural logarithm of both sides. The correct approach involves applying the logarithmic identity for products, leading to ln(6) + 2x = x ln(4). The solution can be simplified to find the value of x, which is given as ln(6) / (ln(4) - 2). This method corrects the initial misunderstanding of moving constants to the exponent.
PREREQUISITES
- Understanding of logarithmic properties, specifically the product rule.
- Familiarity with exponential functions and their manipulation.
- Knowledge of solving equations involving natural logarithms.
- Basic algebra skills for isolating variables.
NEXT STEPS
- Study the properties of logarithms, focusing on the product, quotient, and power rules.
- Practice solving exponential equations using natural logarithms.
- Explore advanced topics in logarithmic functions, such as change of base formula.
- Review algebraic techniques for isolating variables in complex equations.
USEFUL FOR
Students preparing for calculus or algebra exams, educators teaching logarithmic functions, and anyone seeking to improve their problem-solving skills in mathematics.