Question Find all the values of $x$ that satisfy the following equation. \[ \log _{2}(x+2)+\log _{2}(x+5)=2 \] Provide your answer below: \[ x= \] FEEDBACK MORE INSTRUCTION SUBM
Solution
Step 1 ：Combine the two logarithms on the left side of the equation into one logarithm using the property of logarithms. The equation then becomes \( \log _{2}[(x+2)(x+5)]=2 \).
Step 2 ：Convert the logarithmic equation into an exponential equation. The base of the logarithm becomes the base of the exponent, the right side of the equation becomes the exponent, and the number inside the logarithm becomes the result of the exponentiation. The equation then becomes \( 2^2=(x+2)(x+5) \).
Step 3 ：Solve this quadratic equation for x, which gives the solutions \( x = -6 \) and \( x = -1 \).
Step 4 ：Check these solutions in the original equation because the logarithm is not defined for negative numbers and zero. If we substitute \( x = -6 \) into the original equation, we get \( \log _{2}(-6+2)+\log _{2}(-6+5) \), which is not defined because the logarithm is not defined for negative numbers. If we substitute \( x = -1 \) into the original equation, we get \( \log _{2}(-1+2)+\log _{2}(-1+5) \), which is defined and equals 2.
Step 5 ：Therefore, the only solution to the equation is \( x = -1 \).
Step 6 ：Final Answer: \( x=\boxed{-1} \)
