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Find the value of [tex][tex]$x$[/tex][/tex] in the equation [tex](15 - 5x) - \log(3x - 2) - 2 = 0[/tex].

Sagot :

Certainly! Let's solve the given equation step by step.

The given equation is:
[tex]\[ (15 - 5x) - \log(3x - 2) - 2 = 0 \][/tex]

### Step 1: Simplify the Equation
First, we simplify the expression within the equation:
[tex]\[ 15 - 5x - \log(3x - 2) - 2 = 0 \][/tex]

Combine constants:
[tex]\[ (15 - 2) - 5x - \log(3x - 2) = 0 \][/tex]
[tex]\[ 13 - 5x - \log(3x - 2) = 0 \][/tex]

### Step 2: Isolate the Logarithmic Function
Next, move the logarithmic term to one side of the equation:
[tex]\[ 13 - 5x = \log(3x - 2) \][/tex]

### Step 3: Exponentiate Both Sides
To eliminate the logarithmic function, we exponentiate both sides with base [tex]\(e\)[/tex]:
[tex]\[ e^{13 - 5x} = 3x - 2 \][/tex]

### Step 4: Solve for [tex]\(x\)[/tex]
Now our equation is:
[tex]\[ e^{13 - 5x} = 3x - 2 \][/tex]

This is a transcendental equation involving [tex]\(x\)[/tex] in both the exponent and base of different terms. Such equations typically do not have a straightforward algebraic solution.

### Step 5: Use the Lambert W Function
The Lambert W function, denoted as [tex]\(W(z)\)[/tex], satisfies [tex]\(Z = W(Z)e^{W(Z)}\)[/tex]. We can manipulate the equation to apply this function.

Rewriting:
[tex]\[ e^{13} \cdot e^{-5x} = 3x - 2 \][/tex]
Let’s set:
[tex]\[ u = -5x \][/tex]

Substituting back into the equation:
[tex]\[ e^{13} \cdot e^{u} = 3\left(\frac{-u}{5}\right) - 2 \][/tex]
Simplifying further:
[tex]\[ e^{13} \cdot e^{u} = -\frac{3u}{5} - 2 \][/tex]

After isolating [tex]\( e^u \)[/tex]:
[tex]\[ e^{13} \cdot e^{u} + 2 = -\frac{3u}{5} \][/tex]

Multiply through by 5:
[tex]\[ 5e^{13}e^u + 10 = -3u \][/tex]

Now, to isolate [tex]\(e^{u}\)[/tex], we have:
[tex]\[ 5e^{u + 13} = -3u - 10 \][/tex]

Express [tex]\(u\)[/tex] as:
[tex]\[ u = \frac{-10 - 3W \left( \frac{5e^{13}}{-3} \right) }{-3} \][/tex]

Thus:
[tex]\[ x = \frac{2}{3} + \frac{1}{5} W\left( \frac{5e^{29/3}}{3} \right) \][/tex]

### Conclusion
Hence, the solution to the equation [tex]\( (15 - 5x) - \log(3x - 2) - 2 = 0 \)[/tex] is given by:
[tex]\[ x = \frac{2}{3} + \frac{1}{5}W\left(\frac{5e^{29/3}}{3}\right) \][/tex]
This solution involves the Lambert W function, which is typically evaluated using numerical methods or special function libraries in mathematical software.