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Use the laws of logarithms to solve\newlinelog⁡2(16x)+log⁡2(x+1)=3+log⁡2(x+6)\log_{2}(16x)+\log_{2}(x+1)=3+\log_{2}(x+6)

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Full solution

Q. Use the laws of logarithms to solve\newlinelog⁡2(16x)+log⁡2(x+1)=3+log⁡2(x+6)\log_{2}(16x)+\log_{2}(x+1)=3+\log_{2}(x+6)
  1. Combine logs using product property: Combine the logs on the left side using the product property of logarithms; log⁡b(m)+log⁡b(n)=log⁡b(m∗n)\log_b(m) + \log_b(n) = \log_b(m*n).
  2. Subtract to isolate logarithmic terms: Subtract log⁡2(x+6)\log_{2}(x+6) from both sides to isolate the logarithmic terms on one side.
  3. Apply quotient property of logarithms: Apply the quotient property of logarithms; log⁡b(m)−log⁡b(n)=log⁡b(mn)\log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right).
  4. Convert to exponential equation: Convert the logarithmic equation to an exponential equation; if log⁡b(m)=n\log_b(m) = n, then bn=mb^n = m.
  5. Simplify exponential expression: Simplify the exponential expression.
  6. Eliminate denominator by multiplying: Multiply both sides by (x+6)(x+6) to eliminate the denominator.
  7. Distribute and expand equation: Distribute and expand both sides of the equation.
  8. Move terms to set quadratic equation to zero: Move all terms to one side to set the quadratic equation to 00.
  9. Combine like terms: Combine like terms.
  10. Divide entire equation by 88: Divide the entire equation by 88 to simplify.
  11. Factor quadratic equation: Factor the quadratic equation.
  12. Set factors equal to zero: Set each factor equal to 00 and solve for xx.
  13. Solve first equation for xx: Solve the first equation for xx.
  14. Solve second equation for xx: Solve the second equation for xx.

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