Answered

What are the solutions to $\log _6\left(x^2+8\right)=1+\log _6(x)$ ?


A$x=1$ and $x=-8$
B$x=2$ and $x=4$
C$x=-2$ and $x=-4$
D$x=-1$ and $x=8$

Verified Answer :

  • Identify the domain \(x > 0\) for the logarithmic functions to be valid.
  • Rewrite the equation as \(\log_6(x^2+8) = \log_6(6x)\) using logarithm properties.
  • Solve the resulting quadratic equation \(x^2 - 6x + 8 = 0\) by factoring into \((x-2)(x-4) = 0\).
  • Conclude that the solutions are \(x=2\) and \(x=4\), which corresponds to option b.
  • The final answer is \(\boxed{b}\).

Explanation

  1. Determining the Domain

The given equation is \(\log_6(x^2+8) = 1 + \log_6(x)\). First, we determine the domain. The logarithm \(\log_6(x)\) is defined only for \(x > 0\). The term \(\log_6(x^2+8)\) is defined for all real \(x\) because \(x^2+8 > 0\) is always satisfied. Thus, the domain of the equation is \(x > 0\).

  1. Simplifying the Equation

We can simplify the right-hand side of the equation using the property \(\log_b(a) + \log_b(c) = \log_b(ac)\) and the fact that \(1 = \log_6(6)\). Substituting this, we get:
\[\log_6(x^2+8) = \log_6(6) + \log_6(x) = \log_6(6x)\]
Since the logarithm is a one-to-one function, we can equate the arguments of the logarithms:
\[x^2+8 = 6x\]

  1. Solving the Quadratic Equation and Identifying the Solution

To solve the resulting quadratic equation \(x^2 - 6x + 8 = 0\), we look for factors of 8 that sum to \(-6\). The numbers \(-2\) and \(-4\) satisfy this condition, as \((-2) \times (-4) = 8\) and \((-2) + (-4) = -6\). Thus, the equation can be factored as:
\[(x-2)(x-4) = 0\]
This gives the potential solutions \(x=2\) and \(x=4\). Both solutions satisfy the domain condition \(x > 0\). Comparing these results with the given options, we identify that option b is the correct answer.

Examples

Logarithmic equations are used in various fields such as chemistry to determine pH levels, or in acoustics to measure sound intensity (decibels). For instance, if you are modeling sound attenuation over distance, you might encounter logarithmic equations where the variable \(x\) represents distance or intensity ratio. Solving these helps in engineering effective noise control measures, ensuring sound levels stay within safe or desired limits.

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