Problem 83 - 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. $$\log (x+4)-\log 2=\log (5 x+1)$$

Step-by-Step Solution

Answer

The exact answer is \(x = 3/4\). In decimal form, the solution is \(x = 0.75\).

Step 1: Apply Logarithm Quotient Rule

Apply the quotient rule for logarithms. The quotient rule states that \(\log_b (a) - \log_b (c) = \log_b (a/c)\). Hence, the equation can be rewritten as: \(\log (x+4)/2 = \log (5x+1)\)

Step 2: Converting Logarithmic Equation to Exponential Equation

Since the bases of the logarithms are the same, the expressions inside the logarithms must also be the same. Hence, this gives us: \((x+4)/2 = (5x+1)\)

Step 3: Solve for x

Clear the fraction by multiplying everything by 2 and solve for \(x\). This gives \(2x + 8 = 10x + 2\), thus simplifying and grouping like terms leaves us with the \(x\) value: \(x = 3/4\). But this will be the potential solution

Step 4: Check if x is Valid

Even if \(x = 3/4\) solves the equation, it must also be in the domain of the original logarithmic expressions. Plugging it back into the original expression shows that \(x = 3/4\) is valid, because all logarithms are positive. If all logarithm expressions were negative, it would be an invalid solution, since logarithms are undefined for negative numbers.

Step 5: Confirm the answer

The final step confirms \(x = 3/4\) as the solution, therefore it is the exact answer. For decimal approximation correct to two decimal places it would be \(x = 0.75\)