Solve.
$4^x=8$
$x=$
$\square$
(Type an integer or a simplified fraction.)
Answer (1)
2 3
Explanation
Rewrite with Common Base We are given the equation 4 x = 8 and we want to solve for x . The key idea here is to express both sides of the equation using the same base. Since both 4 and 8 are powers of 2, we can rewrite the equation in terms of base 2.
Expressing as Powers of 2 We can rewrite 4 as 2 2 and 8 as 2 3 . Substituting these into the original equation, we get ( 2 2 ) x = 2 3 .
Simplifying the Equation Using the power of a power rule, which states that ( a m ) n = a mn , we can simplify the left side of the equation to 2 2 x = 2 3 .
Equating Exponents Now that the bases are the same, we can equate the exponents. This gives us the equation 2 x = 3 .
Solving for x To solve for x , we divide both sides of the equation by 2: x = 2 3 .
Final Answer Therefore, the solution to the equation 4 x = 8 is x = 2 3 .
Examples
Exponential equations like 4 x = 8 are useful in modeling various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if a population doubles every t years, the population at time x can be modeled as P ( x ) = P 0 ⋅ 2 x / t , where P 0 is the initial population. Solving exponential equations helps us determine the time it takes for a population to reach a certain size or the half-life of a radioactive substance.
