https://file.aiquickdraw.com/mathbot/file/172956441453675ax0t4d.jpg

The equation presented in your image is:

\[
\log(x + 7) = \log x + \log 7
\]

To solve for \( x \), we can use logarithmic properties.

### Step-by-step Solution:
1. **Apply the Product Rule of Logarithms**:  
   The equation \(\log x + \log 7\) can be combined using the product rule:
   \[
   \log x + \log 7 = \log (x \cdot 7)
   \]
   Now, the equation becomes:
   \[
   \log(x + 7) = \log(7x)
   \]

2. **Remove the Logs**:  
   Since \(\log A = \log B\) implies \( A = B \), we can remove the logarithms and set the arguments equal to each other:
   \[
   x + 7 = 7x
   \]

3. **Solve the Linear Equation**:
   Now, solve for \( x \):
   \[
   x + 7 = 7x
   \]
   Subtract \( x \) from both sides:
   \[
   7 = 6x
   \]
   Divide both sides by 6:
   \[
   x = \frac{7}{6}
   \]

So, the value of \( x \) is \( \frac{7}{6} \) or approximately \( 1.1667 \) when rounded to four decimal places.

---

Do you have any further questions, or would you like more details on any part of the solution?

Here are 5 related questions you can explore:
1. What is the product rule for logarithms, and how is it applied in various problems?
2. How do you solve logarithmic equations when the logarithms have different bases?
3. What happens when a logarithmic equation has no solution (e.g., negative arguments)?
4. Can logarithmic functions be used to model real-world exponential growth or decay situations?
5. How do the properties of logarithms simplify complex logarithmic expressions?

**Tip**: Always check the domain of logarithmic equations—logarithms are only defined for positive arguments!

For what value of x is the following true? log(x + 7) = log x + log 7

Learn how to solve the logarithmic equation log(x + 7) = log x + log 7 using the product rule of logarithms. This problem covers key algebraic concepts and logarithmic properties. 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