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Chapter 3: Problem 60

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\log _{2} x+\log _{2}(x+2)=\log _{2}(x+6)$$

Short Answer

Expert verified
The solution for the logarithmic equation is x = 2.

Step by step solution

01

Use the Law of Logarithms

Combine the logarithms on the left side of the equation by using the product rule of logarithms, which states that \( \log_b mn = \log_b m + \log_b n \). So, the equation becomes \( \log _2 (x * (x+2)) = \log _2 (x+6) \)
02

Remove the Logs

Since we have logs of the same base on both sides of the equation, we can drop the logs and get down to the algebraic equation \( x * (x+2) = x+6 \).
03

Simplify the Equation

Simplify the equation by expanding the left-hand side to get \( x^2 + 2x = x+6 \). Then, move everything to one side of the equation to get \( x^2 + x - 6 = 0 \).
04

Solve the Quadratic Equation

Now we have a quadratic equation. We can solve it by factoring. This equation factors to \( (x-2)(x+3)=0 \). Setting each term equal to zero gives the solutions \( x=2, -3 \)
05

Check Validity of Solution

Remember that you can't take the log of a negative number. Therefore, \( x = -3 \) is an extraneous solution and we discard it. So, the only solution to the original equation is \( x = 2 \)

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