Problem 79 - College Algebra Essentials

Question

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\log (3 x-3)=\log (x+1)+\log 4$$

Step-by-Step Solution

Answer

The solution \(x=-7\) doesn't satisfy the original logarithmic equation and is not in the domain of the original logarithmic expressions. Therefore, the given logarithmic equation has no solutions.

Step 1: Combine the right side of the equation

Since we know that the sum of two logarithms equals the logarithm of the product of their arguments, we can apply this property to combine the two logarithms on the right-hand side of our equation. Using the logarithmic property \( \log a + \log b = \log (ab) \) we get: \n\n\( \log (3x - 3) = \log [(x+1) * 4] \)

Step 2: Simplify the equation

Next, we simplify the right side of the equation by multiplying the arguments together to get: \n\n\( \log (3x - 3) = \log (4x + 4) \)

Step 3: Solve for \( x \)

Since the base of the logarithm on both sides of the equation is the same, we can equate their arguments to make the equation simpler. This gives us the equation: \n\n\( 3x - 3 = 4x + 4 \). Solution of this linear equation results in \( x = -7 \)

Step 4: Validate the solution

We must verify if \( x = -7 \) satisfies the original equation and does not lead to taking a logarithm of a negative number. Logarithms of negative numbers are undefined. Substituting \( x = -7 \) into the original equation yields \( \log(-21) = \log(-3) + \log 4 \), so the solution is not valid because it does not satisfy the original equation.