Solve the equation.
[tex]
\begin{array}{l}
2^{x-3}=16 \\
x=[?]
\end{array}
[/tex]
Answer (2)
Rewrite 16 as a power of 2: 16 = 2 4 .
Set the exponents equal: x − 3 = 4 .
Solve for x : x = 4 + 3 .
The solution is: 7 .
Explanation
Understanding the problem We are given the equation 2 x − 3 = 16 and we need to solve for x .
Rewriting the equation First, we can rewrite 16 as a power of 2. Since 16 = 2 × 2 × 2 × 2 = 2 4 , we can rewrite the equation as 2 x − 3 = 2 4 .
Equating the exponents Now that the bases are the same, we can set the exponents equal to each other: x − 3 = 4 .
Solving for x To solve for x , we add 3 to both sides of the equation: x = 4 + 3 .
Final answer Therefore, x = 7 .
Examples
Exponential equations like this are used in various fields, such as calculating population growth, radioactive decay, and compound interest. For example, if a population doubles every 3 years and starts at 1000, we can use an exponential equation to find the population after a certain number of years. Similarly, in finance, compound interest calculations rely on exponential growth to determine the future value of an investment.
The solution to the equation 2 x − 3 = 16 is x = 7 . We achieve this by rewriting 16 as 2 4 and equating the exponents. Therefore, the final answer is x = 7 .
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