Let's solve the equation [tex]\(\log x^2 + \log x = 9\)[/tex] step by step.
1. Apply the properties of logarithms: Recall the property that allows us to combine the sum of logarithms into a single logarithm:
[tex]\[
\log a + \log b = \log(ab)
\][/tex]
Applying this property to the given equation, we have:
[tex]\[
\log x^2 + \log x = \log(x^2 \cdot x) = \log(x^3)
\][/tex]
2. Rewrite the equation: Now the equation becomes:
[tex]\[
\log(x^3) = 9
\][/tex]
3. Remove the logarithm: To eliminate the logarithm, we exponentiate both sides of the equation with base 10 (since the logarithm is in base 10 by default):
[tex]\[
x^3 = 10^9
\][/tex]
4. Solve for [tex]\(x\)[/tex]: To isolate [tex]\(x\)[/tex], take the cube root of both sides of the equation:
[tex]\[
x = \sqrt[3]{10^9}
\][/tex]
5. Calculate the cube root: We know that [tex]\(10^9\)[/tex] represents [tex]\(10\)[/tex] raised to the power of [tex]\(9\)[/tex], so taking the cube root simplifies this to:
[tex]\[
x = 10^{9/3}
\][/tex]
6. Simplify the exponent: Calculate the exponent:
[tex]\[
x = 10^3
\][/tex]
7. Find the numerical value: Hence,
[tex]\[
x = 1000
\][/tex]
Therefore, the solution to the equation [tex]\(\log x^2 + \log x = 9\)[/tex] is:
[tex]\[
x = 1000
\][/tex]
