Problem 84 - 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. $$\log (x+7)-\log 3=\log (7 x+1)$$

Step-by-Step Solution

Answer

The solution to the given logarithmic equation, rounded to two decimal places, is \( x = 0.2 \).

Step 1: Apply the rules of logarithms

The starting point of this problem is the expression \( \log (x+7) - \log 3 = \log (7x+1) \). Using the rules of logarithms, the difference of logs on the left-hand side of the equation can be re-written as a division within a single logarithm: \( \log\left(\frac{x+7}{3}\right) = \log (7x+1) \)

Step 2: Take care of the logarithm

The base of the logarithms is not explicitly stated, but in this form, it is understood to be 10 in this context. Since \( \log_{b}{a} = \log_{b}{c} \) if and only if \( a = c \), the expression inside each logarithm must be equal to each other, yielding the equation \( \frac{x+7}{3} = 7x+1 \)

Step 3: Solve the equation

Manipulate this equation to solve for \( x \). Begin by multiplying through by 3 to eliminate the fraction, yielding \( x+7 = 21x +3 \). It can be further simplified by subtracting \( x \) from both sides to get \( 7 = 20x +3 \). Now, subtract 3 from both sides to get \( 4 = 20x \). To solve for \( x \), divide by 20 on both sides to get \( x = 0.2 \).

Step 4: Validate the solution

Substitute the obtained value of \( x = 0.2 \) back into the original logarithmic expressions to see if they are valid (i.e., that their arguments are greater than 0). Checking the expressions: \( \log(x+7) \), \( \log(3) \), and \( \log(7x + 1) \), you can confirm that each one gets a positive value when \( x = 0.2 \) is used. Hence, the obtained solution is valid.