To solve the equation [tex]\(\log(2x - 100) = 3\)[/tex], we can follow these steps:
1. Understand the logarithmic equation: The given equation is in logarithmic form, [tex]\(\log(2x - 100) = 3\)[/tex]. To solve for [tex]\(x\)[/tex], we'll convert this logarithmic form into its exponential form.
2. Convert to exponential form: Recall that [tex]\(\log_b(a) = c\)[/tex] is equivalent to [tex]\(a = b^c\)[/tex]. In this problem, the base of the logarithm is assumed to be 10 (common logarithm). Thus, we can write:
[tex]\[
2x - 100 = 10^3
\][/tex]
3. Simplify the exponential equation: We know that [tex]\(10^3 = 1000\)[/tex]. Substituting this into the equation, we get:
[tex]\[
2x - 100 = 1000
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
- Add 100 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
2x = 1000 + 100
\][/tex]
- Simplify the right-hand side:
[tex]\[
2x = 1100
\][/tex]
- Divide both sides by 2 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{1100}{2} = 550
\][/tex]
Therefore, the solution to the equation [tex]\(\log(2x - 100) = 3\)[/tex] is [tex]\(x = 550\)[/tex].
The correct answer is:
B. [tex]\(x = 550\)[/tex]
