Problem 87 - 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. $$\log x+\log (x+3)=\log 10$$

Step-by-Step Solution

Answer

The solution to the logarithmic equation is \(x = 2\).

Step 1: Use the logs properties

Consolidate the left side of the equation using the logarithmic property \(\log_a(b) + \log_a(c) = \log_a(bc)\), which results in: \(\log (x^2+3x) = \log 10\)

Step 2: Equate the arguments

Since the base and the logs on both sides of the equation are the same, set the arguments equal to each other: \(x^2 + 3x = 10\).

Step 3: Solve the quadratic equation

To solve the quadratic equation, first rearrange the equation to equals 0: \(x^2 + 3x - 10 = 0\). This equations can be factored to \((x - 2)(x + 5) = 0\), which gives solutions \(x = 2\) and \(x = -5\).

Step 4: Verify the solutions

Check whether both solutions are part of the domain of the original logarithmic expression (which included \(\log x\) and \(\log (x+3)\)). \(x = 2\) is allowed since it results in logarithms of positive numbers. But \(x = -5\) is not allowed as \(x+3\) would equal to -2, and we cannot take the logarithm of a negative number. Hence, remove \(x = -5\) from the possible solutions.