Solve for $x$ in this equation:
$4^{5 x}=\left(\frac{1}{32}\right)^{1-x}$
Answer (2)
To solve the equation 4 5 x = ( 32 1 ) 1 − x , we rewrote it using powers of 2 and simplified. After equating the exponents and solving the resulting equation, we found x = − 1 .
;
Rewrite the equation using powers of 2: ( 2 2 ) 5 x = ( 2 − 5 ) 1 − x .
Simplify the exponents: 2 10 x = 2 − 5 ( 1 − x ) .
Equate the exponents: 10 x = − 5 ( 1 − x ) .
Solve for x : x = − 1 . The final answer is − 1 .
Explanation
Problem Analysis We are given the equation 4 5 x = ( 32 1 ) 1 − x . Our goal is to solve for x .
Rewriting with Powers of 2 First, we rewrite the equation using powers of 2. Since 4 = 2 2 and 32 = 2 5 , we can rewrite the equation as ( 2 2 ) 5 x = ( 2 − 5 ) 1 − x .
Simplifying Exponents Next, we simplify the exponents. Using the power of a power rule, we have 2 10 x = 2 − 5 ( 1 − x ) .
Equating Exponents Since the bases are equal, we can equate the exponents: 10 x = − 5 ( 1 − x ) .
Solving for x Now, we solve the linear equation for x . Expanding the right side, we get 10 x = − 5 + 5 x .
Isolating x Isolating x , we subtract 5 x from both sides: 10 x − 5 x = − 5 , which simplifies to 5 x = − 5 .
Finding the Value of x Finally, we solve for x by dividing both sides by 5: x = 5 − 5 = − 1 .
Final Answer Therefore, the solution is x = − 1 .
Examples
Exponential equations are used in various fields such as finance, physics, and computer science. For example, they can model population growth, radioactive decay, and the performance of algorithms. Understanding how to solve exponential equations is crucial for making predictions and analyzing data in these fields.
