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]
[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]
