To determine between which two integers [tex]\(x\)[/tex] lies for the equation [tex]\(2^x = 26\)[/tex], we can follow these steps:
1. Understand the problem: We need to find [tex]\(x\)[/tex] such that [tex]\(2^x = 26\)[/tex], and then identify the two integers between which [tex]\(x\)[/tex] falls.
2. Rewrite the equation: We start with the given equation:
[tex]\[
2^x = 26
\][/tex]
3. Apply logarithms: To isolate [tex]\(x\)[/tex], we take the logarithm of both sides of the equation. For simplicity, we use the base-2 logarithm:
[tex]\[
\log_2(2^x) = \log_2(26)
\][/tex]
4. Simplify using the properties of logarithms: We know that [tex]\(\log_b(b^y) = y\)[/tex], so the left-hand side simplifies to [tex]\(x\)[/tex]:
[tex]\[
x = \log_2(26)
\][/tex]
5. Determine the numerical value: After computing [tex]\(\log_2(26)\)[/tex], the approximate value of [tex]\(x\)[/tex] is:
[tex]\[
x \approx 4.7004
\][/tex]
6. Identify the integer bounds: We now need to identify the two closest integers between which [tex]\(x\)[/tex] falls. The value [tex]\(4.7004\)[/tex] lies between the integers 4 and 5.
Thus, [tex]\(x\)[/tex] falls between the integers 4 and 5.
Conclusion: The value of [tex]\(x\)[/tex] for the equation [tex]\(2^x = 26\)[/tex] is approximately 4.7004, which lies between the integers 4 and 5. Therefore, [tex]\(x\)[/tex] is between 4 and 5.
