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Chapter 3: Problem 108

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
Solving this logarithmic equation algebraically, the correct solution is approximately \(x = 2.000\).

Step by step solution

01

Combine the Logarithms

Use the logarithm product property \( \log_b(M) + \log_b(N) = \log_b(MN) \) to combine the left-hand side logs: \( \log_2(x(x+2)) = \log_2(x+6) \).
02

Convert the Logarithm to Exponential Form

That is, \( \text{if } \log_b(A) = N, \text{ then } b^N = A \). Using this, you get \( x(x+2) = x+6 \).
03

Solve the Resulting Quadratic Equation

Rewrite the equation in the standard quadratic form: \( x^2 + 2x = x+6 \). After doing this, simplify to \( x^2 + x - 6 = 0 \). Solve this equation, getting \(x = 2\) or \(x = -3\)
04

Eliminate Extraneous Solutions

Test the solutions of the equation in the original logarithm. Logarithms only accept positive arguments, so if a solution makes the argument negative, it is rejected. In this case, \(x=-3\) is an extraneous solution because it makes the arguments \(x\) and \(x+2\) negative in the original equation. Thus, a single solution remains: \(x=2\).
05

Approximate the Result

Because of the nature of this problem, approximation isn't necessary as the result is already exact. However, if you wish to round to three decimal places, the solution would still be \(x = 2.000\).

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Most popular questions from this chapter

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$4 \log (x-6)=11$$

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