Let's simplify the given equation step by step:

The equation is:

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

### Step 1: Simplify \((10^{-3})^0\)
Any non-zero number raised to the power of 0 is equal to 1. Therefore:

\[
(10^{-3})^0 = 1
\]

Now the equation becomes:

\[
\log_x(1000) - \log_x(x^2) + 1 = 0
\]

### Step 2: Apply the logarithmic properties

Using the property \(\log_b(a^n) = n \log_b(a)\), we simplify \(\log_x(x^2)\):

\[
\log_x(x^2) = 2 \log_x(x)
\]

Since \(\log_x(x) = 1\), we get:

\[
\log_x(x^2) = 2
\]

Now the equation is:

\[
\log_x(1000) - 2 + 1 = 0
\]

Simplifying further:

\[
\log_x(1000) - 1 = 0
\]

### Step 3: Solve for \(\log_x(1000)\)

Rearranging:

\[
\log_x(1000) = 1
\]

This means \(x^1 = 1000\), so \(x = 1000\).

### Final Answer:
The solution to the equation is:

\[
x = 1000
\]

Would you like more details or have any other questions?

Here are 5 related questions to explore further:
1. What are the key logarithmic properties used in this problem?
2. How do you solve equations involving different logarithmic bases?
3. What is the significance of the equation \(\log_x(x^n)\)?
4. How does the base of a logarithm affect the output value?
5. Can the solution change if the equation involves natural logarithms?

**Tip:** Always remember that \(\log_x(x) = 1\) for any base \(x > 0\).

Learn how to solve the logarithmic equation log_x(1000) - log_x(x^2) + (10^(-3))^0 = 0 with a step-by-step breakdown. This solution covers key logarithmic properties, such as log_b(a^n) and the identity log_b(b) = 1, and simplifies the equation to find x = 1000. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation