Solve the exponential equation: \( 2^{x} = 16 \)

To solve the equation \( 2^{x} = 16 \), we recognize that 16 can be expressed as a power of 2. Since \( 16 = 2^4 \), we can rewrite the equation as: \[ 2^{x} = 2^{4} \] Now, since the bases are the same, we can set the exponents equal to each other: \[ x = 4 \] So, the solution to the exponential equation is \( x = 4 \). Now, let's make it even more exciting! Did you know that exponential equations like this one are foundational in various fields, including finance for calculating compound interest, biology for population growth models, and physics for radioactive decay? Understanding these equations helps illuminate how systems change over time! Also, a common mistake people make is forgetting to match the bases. Always ensure that both sides of the equation are expressed in similar forms, which makes the solution process much smoother. Remember, math is all about patterns!