Answer :
To solve the equation [tex]\( x = \lg \left(\frac{1}{3-2x}\right) \)[/tex], let's begin by understanding and transforming the given equation step-by-step.
### Step 1: Rewrite the Logarithmic Equation
The given equation is:
[tex]\[ x = \lg \left(\frac{1}{3-2x}\right) \][/tex]
We know that for any [tex]\( a > 0 \)[/tex] and [tex]\( a \neq 1 \)[/tex]:
[tex]\[ \lg \left(\frac{1}{a}\right) = -\lg(a) \][/tex]
So, applying this property to our equation:
[tex]\[ x = -\lg(3 - 2x) \][/tex]
### Step 2: Express in Terms of Logarithmic Equality
By rearranging, we can rewrite the equation as:
[tex]\[ x = -\lg(3 - 2x) \][/tex]
### Step 3: Introduce a Suitable Straight Line
Notice that [tex]\( y = \lg(a) \)[/tex] can be expressed as [tex]\( a = 10^y \)[/tex]. Trying to find a linear relationship, consider the transformation:
[tex]\[ y = \lg(3 - 2x) \Rightarrow 3 - 2x = 10^y \][/tex]
### Step 4: Formulate a Linear Equation
From the expression [tex]\( 3 - 2x = 10^y \)[/tex], let's isolate [tex]\( y \)[/tex]:
[tex]\[ y = -x \][/tex]
Thus, the equation of the suitable straight line becomes:
[tex]\[ y = -x \][/tex]
### Step 5: Compare with Another Function
We also need to consider the function [tex]\( y = \frac{1}{10^x} \)[/tex].
### Step 6: Intersection of Functions
To find the number of solutions, we look for the intersection points between [tex]\( y = -x \)[/tex] and [tex]\( y = \frac{1}{10^x} \)[/tex].
Equate the two equations:
[tex]\[ -x = \frac{1}{10^x} \][/tex]
### Step 7: Solve the Equation
Multiply both sides by [tex]\( 10^x \)[/tex] to clear the denominator:
[tex]\[ -x \cdot 10^x = 1 \][/tex]
### Step 8: Analyze the Equation
The function [tex]\( -x \cdot 10^x \)[/tex] is monotonic for [tex]\( x > 0 \)[/tex] (it decreases rapidly) and [tex]\( x < 0 \)[/tex] (it decreases rapidly as well). Therefore, examine the behavior only in the critical zones.
### Numerical or Graphical Solution:
This transcendental equation does not have a straightforward algebraic solution. Let's analyze it graphically or numerically.
Use the nature of the functions:
- [tex]\( y = -x \)[/tex] is a linear function.
- [tex]\( y = \frac{1}{10^x} \)[/tex] intersects with [tex]\( y = -x \)[/tex] at some [tex]\( x \)[/tex] value.
By numerical approximation, we consider solving the equation:
[tex]\[ -x \cdot 10^x = 1 \][/tex]
To find the number of solutions:
- Both [tex]\( y = \frac{1}{10^x} \)[/tex] and [tex]\( y = -x \)[/tex] will intersect exactly once, because [tex]\( \frac{1}{10^x} \to 0 \)[/tex] as [tex]\( x \to \infty \)[/tex] and [tex]\( y = -x \to \infty \)[/tex] as [tex]\( x \to -\infty \)[/tex].
Therefore, there is exactly one solution to the equation.
### Conclusion:
- Equation of the straight line: [tex]\( y = -x \)[/tex]
- Number of solutions: One
### Step 1: Rewrite the Logarithmic Equation
The given equation is:
[tex]\[ x = \lg \left(\frac{1}{3-2x}\right) \][/tex]
We know that for any [tex]\( a > 0 \)[/tex] and [tex]\( a \neq 1 \)[/tex]:
[tex]\[ \lg \left(\frac{1}{a}\right) = -\lg(a) \][/tex]
So, applying this property to our equation:
[tex]\[ x = -\lg(3 - 2x) \][/tex]
### Step 2: Express in Terms of Logarithmic Equality
By rearranging, we can rewrite the equation as:
[tex]\[ x = -\lg(3 - 2x) \][/tex]
### Step 3: Introduce a Suitable Straight Line
Notice that [tex]\( y = \lg(a) \)[/tex] can be expressed as [tex]\( a = 10^y \)[/tex]. Trying to find a linear relationship, consider the transformation:
[tex]\[ y = \lg(3 - 2x) \Rightarrow 3 - 2x = 10^y \][/tex]
### Step 4: Formulate a Linear Equation
From the expression [tex]\( 3 - 2x = 10^y \)[/tex], let's isolate [tex]\( y \)[/tex]:
[tex]\[ y = -x \][/tex]
Thus, the equation of the suitable straight line becomes:
[tex]\[ y = -x \][/tex]
### Step 5: Compare with Another Function
We also need to consider the function [tex]\( y = \frac{1}{10^x} \)[/tex].
### Step 6: Intersection of Functions
To find the number of solutions, we look for the intersection points between [tex]\( y = -x \)[/tex] and [tex]\( y = \frac{1}{10^x} \)[/tex].
Equate the two equations:
[tex]\[ -x = \frac{1}{10^x} \][/tex]
### Step 7: Solve the Equation
Multiply both sides by [tex]\( 10^x \)[/tex] to clear the denominator:
[tex]\[ -x \cdot 10^x = 1 \][/tex]
### Step 8: Analyze the Equation
The function [tex]\( -x \cdot 10^x \)[/tex] is monotonic for [tex]\( x > 0 \)[/tex] (it decreases rapidly) and [tex]\( x < 0 \)[/tex] (it decreases rapidly as well). Therefore, examine the behavior only in the critical zones.
### Numerical or Graphical Solution:
This transcendental equation does not have a straightforward algebraic solution. Let's analyze it graphically or numerically.
Use the nature of the functions:
- [tex]\( y = -x \)[/tex] is a linear function.
- [tex]\( y = \frac{1}{10^x} \)[/tex] intersects with [tex]\( y = -x \)[/tex] at some [tex]\( x \)[/tex] value.
By numerical approximation, we consider solving the equation:
[tex]\[ -x \cdot 10^x = 1 \][/tex]
To find the number of solutions:
- Both [tex]\( y = \frac{1}{10^x} \)[/tex] and [tex]\( y = -x \)[/tex] will intersect exactly once, because [tex]\( \frac{1}{10^x} \to 0 \)[/tex] as [tex]\( x \to \infty \)[/tex] and [tex]\( y = -x \to \infty \)[/tex] as [tex]\( x \to -\infty \)[/tex].
Therefore, there is exactly one solution to the equation.
### Conclusion:
- Equation of the straight line: [tex]\( y = -x \)[/tex]
- Number of solutions: One
