Solve for \( x \) : \( \log (x+5)-\log (x+4)=1 \) \( x=\square \) You may enter the exact value or round to 4 decimal places.

**Step 1: Combine the logarithms using the quotient rule** We start with the given equation: \[ \log (x+5) - \log (x+4) = 1 \] Using the logarithm quotient rule, we rewrite the left-hand side as: \[ \log \left(\frac{x+5}{x+4}\right) = 1 \] **Step 2: Rewrite the logarithmic equation in exponential form** Assuming the logarithm is base 10, the equation in exponential form becomes: \[ \frac{x+5}{x+4} = 10^1 = 10 \] **Step 3: Solve for \( x \)** Multiply both sides of the equation by \( x+4 \): \[ x+5 = 10(x+4) \] Expand the right-hand side: \[ x+5 = 10x + 40 \] Rearrange the equation to isolate \( x \): \[ 5 - 40 = 10x - x \] \[ -35 = 9x \] Divide both sides by 9: \[ x = -\frac{35}{9} \] **Step 4: Check the domain restrictions** For the logarithms to be defined, the arguments must be positive: \[ x+5 > 0 \quad \text{and} \quad x+4 > 0 \] Substitute \( x = -\frac{35}{9} \) into the inequalities: \[ -\frac{35}{9} + 5 = \frac{10}{9} > 0 \] \[ -\frac{35}{9} + 4 = \frac{1}{9} > 0 \] Both conditions are satisfied. **Answer** \[ x = -\frac{35}{9} \]