https://file.aiquickdraw.com/mathbot/file/1725712136656jee6ex0q.jpg

The equation shown in the image is:

\[
\log_x 1000 - \log_x x^2 + (10^{-3})^0 = 0
\]

Let's break this down step by step.

1. **Simplify \((10^{-3})^0\):** 
   \[
   (10^{-3})^0 = 1 \quad \text{(any non-zero number raised to the power of 0 equals 1)}.
   \]
   So the equation becomes:
   \[
   \log_x 1000 - \log_x x^2 + 1 = 0.
   \]

2. **Use the logarithmic property**:
   \[
   \log_x x^2 = 2 \log_x x = 2.
   \]
   Now substitute this into the equation:
   \[
   \log_x 1000 - 2 + 1 = 0.
   \]
   Simplifying further:
   \[
   \log_x 1000 - 1 = 0.
   \]

3. **Solve for \(\log_x 1000\):**
   \[
   \log_x 1000 = 1.
   \]
   By the definition of logarithms, this means:
   \[
   x^1 = 1000,
   \]
   so:
   \[
   x = 1000.
   \]

Thus, the solution to the equation is \(x = 1000\).

Would you like a deeper explanation of any step or have any questions?

Here are 5 related questions you might find interesting:
1. What are the general properties of logarithms?
2. How can logarithmic functions be inverted?
3. What happens when the logarithm base is a fraction?
4. How do you solve logarithmic equations with different bases?
5. How is the concept of logarithms used in real-life applications?

**Tip:** Always try simplifying terms like \((10^{-3})^0\) or logarithmic expressions as a first step—it often makes equations much easier to manage!

In this problem, we solve a logarithmic equation using the properties of logarithms and exponentiation. The equation includes log_x 1000 - log_x x^2 + (10^{-3})^0 = 0. We apply logarithmic properties and simplifications, resulting in x = 1000. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation