The equation `(log_(x) 10)^(3) -(log_(x) 10)^(2) - 6 log_(x) 10 = 0 ` is
satisfied by a value of `x` given by

Options

A. `root(3)(10)`
B. `(1)/(sqrt(10))`
C. `sqrt(10)`
D. `(1)/(root(3)(10))`

Answer

To solve the equation \((\log_{x} 10)^{3} - (\log_{x} 10)^{2} - 6 \log_{x} 10 = 0\), we will follow these steps:

### Step 1: Substitute the logarithm
Let \( a = \log_{x} 10 \). Then, the equation can be rewritten as:
\[ a^{3} - a^{2} - 6a = 0 \]

### Step 2: Factor out \( a \)
We can factor out \( a \) from the equation:
\[ a(a^{2} - a - 6) = 0 \]
This gives us two cases to consider: \( a = 0 \) or \( a^{2} - a - 6 = 0 \).

### Step 3: Solve the quadratic equation
Now, we will solve the quadratic equation \( a^{2} - a - 6 = 0 \) using the middle-term splitting method:
\[ a^{2} - 3a + 2a - 6 = 0 \]
Grouping the terms, we have:
\[ a(a - 3) + 2(a - 3) = 0 \]
Factoring gives:
\[ (a - 3)(a + 2) = 0 \]
Thus, we have two solutions for \( a \):
\[ a - 3 = 0 \quad \Rightarrow \quad a = 3 \]
\[ a + 2 = 0 \quad \Rightarrow \quad a = -2 \]

### Step 4: Convert back to logarithmic form
Now, we will convert back to logarithmic form for both values of \( a \).

1. For \( a = 3 \):
\[ \log_{x} 10 = 3 \quad \Rightarrow \quad \frac{\log 10}{\log x} = 3 \]
This implies:
\[ \log 10 = 3 \log x \quad \Rightarrow \quad \log 10 = \log x^{3} \]
Taking the antilogarithm:
\[ 10 = x^{3} \quad \Rightarrow \quad x = 10^{1/3} \]

2. For \( a = -2 \):
\[ \log_{x} 10 = -2 \quad \Rightarrow \quad \frac{\log 10}{\log x} = -2 \]
This implies:
\[ \log 10 = -2 \log x \quad \Rightarrow \quad \log 10 = \log x^{-2} \]
Taking the antilogarithm:
\[ 10 = x^{-2} \quad \Rightarrow \quad x^{2} = \frac{1}{10} \quad \Rightarrow \quad x = 10^{-1/2} = \frac{1}{\sqrt{10}} \]

### Final Solutions
The values of \( x \) that satisfy the original equation are:
\[ x = 10^{1/3} \quad \text{and} \quad x = \frac{1}{\sqrt{10}} \]