To solve the equation [tex]\(2^x + 1 = 9\)[/tex] for [tex]\(x\)[/tex], follow these steps:
1. Isolate [tex]\(2^x\)[/tex]:
Start by isolating the term with the exponent, which in this case is [tex]\(2^x\)[/tex]. We do this by subtracting 1 from both sides of the equation:
[tex]\[
2^x + 1 - 1 = 9 - 1
\][/tex]
Simplifying, we get:
[tex]\[
2^x = 8
\][/tex]
2. Apply logarithms:
To solve for [tex]\(x\)[/tex], we need to get rid of the exponent. We can do this by taking the logarithm base 2 of both sides of the equation:
[tex]\[
\log_2 (2^x) = \log_2 (8)
\][/tex]
3. Simplify using logarithmic properties:
Use the property of logarithms that states [tex]\(\log_b (a^c) = c \cdot \log_b (a)\)[/tex]. Applying this property, we get:
[tex]\[
x \cdot \log_2 (2) = \log_2 (8)
\][/tex]
4. Simplify further:
Since [tex]\(\log_2 (2) = 1\)[/tex], the equation simplifies to:
[tex]\[
x \cdot 1 = \log_2 (8)
\][/tex]
So we have:
[tex]\[
x = \log_2 (8)
\][/tex]
5. Calculate [tex]\(\log_2 (8)\)[/tex] using the change of base formula:
The change of base formula states that [tex]\(\log_b (a) = \frac{\log_c (a)}{\log_c (b)}\)[/tex], where [tex]\(c\)[/tex] is any positive number. Using base 10 for simplicity:
[tex]\[
\log_2 (8) = \frac{\log_{10} (8)}{\log_{10} (2)}
\][/tex]
6. Evaluate the logarithms:
Using common logarithms:
[tex]\[
\log_{10} (8) \approx 0.903
\][/tex]
[tex]\[
\log_{10} (2) \approx 0.301
\][/tex]
Dividing these values, we get:
[tex]\[
\log_2 (8) = \frac{0.903}{0.301} \approx 3
\][/tex]
Thus, the solution for [tex]\(x\)[/tex] is:
[tex]\[
x \approx 3.0
\][/tex]
So, rounded to the nearest thousandth, the value of [tex]\(x\)[/tex] is [tex]\(\boxed{3.000}\)[/tex].
