Questions: Solve the following logarithmic equation. ln (x+8)+ln (x-5)=2 ln x

Solution Steps

The given equation is:

\[ \ln (x+8) + \ln (x-5) = 2 \ln x \]

We can use the property of logarithms that states \(\ln a + \ln b = \ln (ab)\). Applying this property to the left side of the equation, we have:

\[ \ln ((x+8)(x-5)) = 2 \ln x \]

The right side of the equation, \(2 \ln x\), can be rewritten using the power rule for logarithms, which states \(a \ln b = \ln (b^a)\). Thus:

\[ 2 \ln x = \ln (x^2) \]

Now the equation becomes:

\[ \ln ((x+8)(x-5)) = \ln (x^2) \]

Since the natural logarithm function is one-to-one, we can exponentiate both sides to eliminate the logarithms:

\[ (x+8)(x-5) = x^2 \]

Expand the left side of the equation:

\[ x^2 + 8x - 5x - 40 = x^2 \]

Simplify the expression:

\[ x^2 + 3x - 40 = x^2 \]

Subtract \(x^2\) from both sides:

\[ 3x - 40 = 0 \]

Add 40 to both sides:

\[ 3x = 40 \]

Divide both sides by 3:

\[ x = \frac{40}{3} \]

We need to ensure that the solution satisfies the original equation and that the arguments of the logarithms are positive. Substitute \(x = \frac{40}{3}\) back into the arguments of the logarithms:

• \(x + 8 = \frac{40}{3} + 8 = \frac{64}{3}\)
• \(x - 5 = \frac{40}{3} - 5 = \frac{25}{3}\)

Both are positive, so the solution is valid.

Final Answer