Answer :

Let's solve the given logarithmic equation step by step:

[tex]\[ \log \left(x^2 - 5\right) - \log x = \log 4 \][/tex]

Step 1: Use the properties of logarithms to combine the logarithmic expressions on the left side of the equation

We can use the logarithmic property [tex]\(\log a - \log b = \log \left(\frac{a}{b}\right)\)[/tex]:

[tex]\[ \log \left(\frac{x^2 - 5}{x}\right) = \log 4 \][/tex]

Step 2: Since the logarithmic functions on both sides of the equation are equal, their arguments must be equal

[tex]\[ \frac{x^2 - 5}{x} = 4 \][/tex]

Step 3: Solve the resulting algebraic equation

Multiply both sides of the equation by [tex]\(x\)[/tex] to eliminate the fraction:

[tex]\[ x^2 - 5 = 4x \][/tex]

Rearrange the equation to set it to zero:

[tex]\[ x^2 - 4x - 5 = 0 \][/tex]

Step 4: Solve the quadratic equation

We can factor the quadratic equation:

[tex]\[ x^2 - 4x - 5 = (x - 5)(x + 1) = 0 \][/tex]

Set each factor equal to zero to find the solutions:

[tex]\[ x - 5 = 0 \quad \text{or} \quad x + 1 = 0 \][/tex]

This gives us:

[tex]\[ x = 5 \quad \text{or} \quad x = -1 \][/tex]

Step 5: Verify the solutions in the context of the original logarithmic equation

Substitute [tex]\(x = 5\)[/tex] back into the original equation to check for validity:

[tex]\[ \log \left(5^2 - 5\right) - \log 5 = \log 4 \][/tex]
[tex]\[ \log \left(25 - 5\right) - \log 5 = \log 4 \][/tex]
[tex]\[ \log 20 - \log 5 = \log 4 \][/tex]
[tex]\[ \log \left(\frac{20}{5}\right) = \log 4 \][/tex]
[tex]\[ \log 4 = \log 4 \quad \text{(True)} \][/tex]

Substitute [tex]\(x = -1\)[/tex] back into the original equation to check for validity:

[tex]\[ \log \left((-1)^2 - 5\right) - \log (-1) = \log 4 \][/tex]
[tex]\[ \log \left(1 - 5\right) - \log (-1) = \log 4 \][/tex]
[tex]\[ \log(-4) - \log(-1) = \log 4 \][/tex]

Notice that [tex]\(\log(-4)\)[/tex] and [tex]\(\log(-1)\)[/tex] are undefined in the real number system because the logarithm of a negative number is not defined for real numbers.

Therefore, the solution [tex]\(x = -1\)[/tex] is extraneous and not valid.

Conclusion:

The only valid solution to the equation [tex]\(\log \left(x^2 - 5\right) - \log x = \log 4\)[/tex] is

[tex]\[ x = 5 \][/tex]