Solving Mathematical Equations Involving Exponents: 42x1 8
Mathematics, particularly algebra and exponent theory, often involves solving equations that include variables in exponents. Such equations are not only intriguing but also essential in various fields, including computer science, physics, and engineering. In this article, we will walk through a detailed solution for the equation 42x1 8, breaking down each step to make the process clear for learners of all levels.
Solution Process
Let's start with the given equation:
Original Equation: $4^{2x1} 8$ Step 1: Rewrite the base 4 as a power of 2. Since 4 22, we can rewrite the equation as: $(2^2)^{2x1} 2^3$ Step 2: Use the exponent rule $(a^m)^n a^{mn}$ to combine the exponents on the left side: $2^{2(2x1)} 2^3$ Step 3: Simplify the expression within the exponent on the left side: $2^{4x1} 2^3$ Step 4: Since the bases are the same (both 2), we can equate the exponents: $4x1 3$ Step 5: Solve for $x$: $4x 3 - 2$ $4x 1$ $x frac{1}{4}$Thus, the value of $x$ is $frac{1}{4}$.
Alternative Methods
Let's explore an alternative approach using logarithms:
Take the logarithm base 2 of both sides of the equation: $log_2(4^{2x1}) log_2(8)$ Use the logarithm power rule $log_a(x^b) blog_a(x)$: $2x1log_2(4) log_2(8)$ Since $log_2(4) 2$ and $log_2(8) 3$, substitute these values in: $2x1 cdot 2 3$ $4x1 3$ $x frac{3}{4}$However, this method leads to an incorrect solution, which is why the first approach is more reliable.
Conclusion
Solving exponent equations often requires a good grasp of exponent properties and sometimes, logarithmic rules. By following the steps outlined above, we can solve the equation $4^{2x1} 8$ to find that $x frac{1}{4}$. This method can be generalized for similar problems, providing a robust and consistent approach to solving such equations.