2
(210g3x) - 4
=10g3（21）一
10g3 (7)
1
10g= 3
Find lowest value of X
- 8

Let's break down the given equation step by step to find the lowest value of \( x \).

The equation appears to be:

\[
2 \cdot (210g3x) - 4 = 10g3(21) - 10g3(7) - 8
\]

### Clarifying the problem
Based on the symbols used, I assume that "g" refers to a constant or possibly a logarithmic function (like \(\log_3\)). I will proceed with both possible interpretations, but you can let me know if another interpretation is more accurate.

#### Scenario 1: Treating "g" as a multiplication constant
1. **Rewrite the equation:**
   \[
   2 \cdot (210 \cdot 3x) - 4 = 10 \cdot 3 \cdot 21 - 10 \cdot 3 \cdot 7 - 8
   \]
   
2. **Simplify both sides:**
   \[
   2 \cdot 630x - 4 = 630 - 210 - 8
   \]
   \[
   1260x - 4 = 412
   \]

3. **Solve for \( x \):**
   \[
   1260x = 416
   \]
   \[
   x = \frac{416}{1260}
   \]
   \[
   x = \frac{104}{315}
   \]

The lowest value of \( x \) in this case is \( \frac{104}{315} \).

#### Scenario 2: Treating "g" as a logarithmic function, i.e., \( \log_3 \)
1. **Rewrite the equation using logarithms:**
   \[
   2 \cdot \log_3(210x) - 4 = \log_3(21) - \log_3(7) - 8
   \]

2. **Simplify the logarithmic expressions:**
   \[
   2 \cdot \log_3(210x) - 4 = \log_3\left(\frac{21}{7}\right) - 8
   \]
   \[
   2 \cdot \log_3(210x) - 4 = \log_3(3) - 8
   \]
   \[
   2 \cdot \log_3(210x) - 4 = 1 - 8
   \]
   \[
   2 \cdot \log_3(210x) - 4 = -7
   \]

3. **Solve for \( \log_3(210x) \):**
   \[
   2 \cdot \log_3(210x) = -3
   \]
   \[
   \log_3(210x) = -\frac{3}{2}
   \]

4. **Convert the logarithmic equation to exponential form:**
   \[
   210x = 3^{-\frac{3}{2}} = \frac{1}{3^{\frac{3}{2}}} = \frac{1}{3 \sqrt{3}}
   \]

5. **Solve for \( x \):**
   \[
   x = \frac{1}{210 \cdot 3 \sqrt{3}}
   \]

The lowest value of \( x \) in this case is \( \frac{1}{210 \cdot 3 \sqrt{3}} \).

### Conclusion
- If "g" is multiplication, \( x = \frac{104}{315} \).
- If "g" is \( \log_3 \), then \( x = \frac{1}{210 \cdot 3 \sqrt{3}} \).

Let me know if you want further clarification on any steps!

#### 5 Related Questions:
1. How do we solve logarithmic equations involving different bases?
2. Can you explain how to solve linear equations with fractions?
3. What is the process of converting between logarithmic and exponential forms?
4. How do we simplify expressions involving large constants?
5. How can we solve logarithms with roots and fractions involved?

**Tip:** When working with logarithms, always double-check the base to avoid confusion in the calculations.

2(210g3x) - 4 = 10g3(21) - 10g3(7) - 8. Find the lowest value of X.

Step-by-step solution to find the lowest value of X for the equation 2(210g3x) - 4 = 10g3(21) - 10g3(7) - 8. This problem covers key concepts in algebra, logarithms, and solving equations, suitable for Grades 10-12 students.

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