To solve the equation [tex]\((-10)^x = 1000\)[/tex], let's break it down step by step.
1. Recognize the properties of the equation:
- The base in our equation is [tex]\(-10\)[/tex], which is negative.
- The exponent [tex]\(x\)[/tex] will determine how the negative base behaves.
- We need to ensure the resulting value, [tex]\(1000\)[/tex], is positive.
2. Consider the behavior of exponents with negative bases:
- For [tex]\(x\)[/tex] to result in a positive value when raising a negative number, [tex]\(x\)[/tex] must be an even integer. However, [tex]\((-10)^x\)[/tex] for even [tex]\(x\)[/tex] produces a positive number, and odd [tex]\(x\)[/tex] produces a negative number. Since we need a positive [tex]\(1000\)[/tex], we focus on [tex]\(10^x = 1000\)[/tex].
3. Disregard the negative sign for now:
- We simplify our problem to study the magnitude only: [tex]\(10^x = 1000\)[/tex].
4. Transform the equation using logarithms:
- To solve for [tex]\(x\)[/tex], we can take the logarithm to base [tex]\(10\)[/tex] of both sides of the simplified equation:
[tex]\[
\log_{10}(10^x) = \log_{10}(1000)
\][/tex]
5. Use properties of logarithms:
- The property [tex]\(\log_{10}(a^b) = b \cdot \log_{10}(a)\)[/tex] helps simplify the left side:
[tex]\[
x \cdot \log_{10}(10) = \log_{10}(1000)
\][/tex]
- Since [tex]\(\log_{10}(10) = 1\)[/tex], this reduces to:
[tex]\[
x = \log_{10}(1000)
\][/tex]
6. Evaluate the logarithm:
- We know that:
[tex]\[
1000 = 10^3
\][/tex]
- Thus,
[tex]\[
\log_{10}(1000) = \log_{10}(10^3) = 3
\][/tex]
Therefore, the solution to the equation [tex]\((-10)^x = 1000\)[/tex] is [tex]\( \boxed{3} \)[/tex].
