For what value of $x$ does $4^x=\left(\frac{1}{8}\right)^{x+5}$?
Answer (1)
Rewrite both sides of the equation with the same base: 4 x = ( 2 2 ) x and ( 8 1 ) x + 5 = ( 2 − 3 ) x + 5 .
Simplify the exponents: 2 2 x = 2 − 3 ( x + 5 ) .
Equate the exponents: 2 x = − 3 ( x + 5 ) .
Solve for x : x = − 3 , so the final answer is − 3 .
Explanation
Understanding the Problem We are given the equation 4 x = ( 8 1 ) x + 5 and we need to find the value of x that satisfies this equation.
Rewriting with the Same Base To solve this equation, we can rewrite both sides with the same base. Notice that 4 = 2 2 and 8 1 = 2 − 3 . Substituting these into the equation, we get ( 2 2 ) x = ( 2 − 3 ) x + 5 .
Simplifying Exponents Now, we simplify the exponents. Using the power of a power rule, we have 2 2 x = 2 − 3 ( x + 5 ) .
Equating Exponents Since the bases are equal, the exponents must be equal. Therefore, we have 2 x = − 3 ( x + 5 ) .
Solving for x Now, we solve the linear equation for x . Expanding the right side, we get 2 x = − 3 x − 15 . Adding 3 x to both sides, we have 5 x = − 15 . Dividing both sides by 5, we find x = − 3 .
Checking the Solution To check our answer, we can substitute x = − 3 back into the original equation: 4 − 3 = ( 8 1 ) − 3 + 5 . This simplifies to 4 − 3 = ( 8 1 ) 2 . We have 4 − 3 = 4 3 1 = 64 1 and ( 8 1 ) 2 = 8 2 1 = 64 1 . Since both sides are equal, our solution is correct.
Final Answer Therefore, the value of x that satisfies the equation is x = − 3 .
Examples
Exponential equations are useful in modeling various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if you invest money in an account that compounds interest, the amount of money you have after a certain time can be modeled using an exponential equation. Similarly, the decay of a radioactive substance can be modeled using an exponential equation. Understanding how to solve exponential equations allows us to make predictions and analyze these phenomena.
