Question

Solve for x:
log(x + 4) -log(x + 1) = 2

Answer 1

The solution to the logarithmic equation
`\( \log(x + 4) - \log(x + 1) = 2 \)` is
`\( x = -(32)/(33) \)`.

To solve the logarithmic equation
`\( \log(x + 4) - \log(x + 1) = 2 \)`, you can use logarithmic properties.

Recall the property
`\( \log(a) - \log(b) = \log\left((a)/(b)\right) \)`. Apply this property to the given equation:
`\[ \log(x + 4) - \log(x + 1) = \log\left((x + 4)/(x + 1)\right) \]`

Now, the equation becomes:
`\[ \log\left((x + 4)/(x + 1)\right) = 2 \]`

To eliminate the logarithm, you can rewrite the equation in exponential form. Since the base of the logarithm is 10 (common logarithm), you can rewrite it as:
`\[ 10^2 = (x + 4)/(x + 1) \]`

Now, solve for x:
`\[ 100 = (x + 4)/(x + 1) \]`

Multiply both sides b (x + 1) to get rid of the fraction: 100(x + 1) = x + 4

Distribute on the left side: 100x + 100 = x + 4

Subtract x from both sides: 99x + 100 = 4

Subtract 100 from both sides: 99x = -96

Divide by 99 to solve for x:
`\[ x = -(96)/(99) \]`

Simplify the fraction:
`\[ x = -(32)/(33) \]`

So, the solution to the equation is
`\( x = -(32)/(33) \)`.