Verified Answer :

  • Determine the domain of the equation: \(t > -2\) and \(t < \frac{14}{3}\).
  • Equate the arguments of the logarithm: \(2t + 4 = 14 - 3t\).
  • Simplify the linear equation: \(5t = 10\) to get \(t = 2\).
  • Verify the solution satisfies the domain and is equivalent to the given choice: \(\boxed{2}\).

Explanation

  1. Domain analysis of the equation

The given equation is \(\log(2t + 4) = \log(14 - 3t)\). For this logarithmic equation to be well-defined, the arguments of the logarithm must be strictly positive. Therefore, we require \(2t + 4 > 0\) and \(14 - 3t > 0\).

  • From the first condition: \(2t > -4\), which simplifies to \(t > -2\).
  • From the second condition: \(14 > 3t\), which simplifies to \(t < \frac{14}{3}\).

Combining these, the valid range for \(t\) is \(-2 < t < \frac{14}{3}\).

  1. Solving the equation for \(t\)

Since the logarithm function is one-to-one, we can equate the arguments of the logarithms on both sides:
\[2t + 4 = 14 - 3t\]
To solve for \(t\), move all terms involving \(t\) to one side and constants to the other:
\[2t + 3t = 14 - 4\]
Combining like terms gives:
\[5t = 10\]
Dividing by 5 results in:
\[t = 2\]

  1. Verification and final conclusion

We must ensure that our calculated solution \(t = 2\) lies within the domain identified in step 1.

  • We check the condition \(-2 < t < \frac{14}{3}\).
  • Substituting \(t = 2\): \(-2 < 2 < \frac{14}{3}\) is approximately \(-2 < 2 < 4.67\), which is true.
  • Verification by substitution into original equation: \(\log(2(2) + 4) = \log(8)\) and \(\log(14 - 3(2)) = \log(8)\).

Since both sides are equal, the solution is correct. Comparing this result with the given choices, the correct option is 2.

Examples

In real-world scenarios, logarithmic equations are used to model phenomena where growth occurs at different rates, such as in chemistry when calculating pH levels or in finance for compound interest rates. For instance, if you were comparing two chemical mixtures where the concentrations of ions follow a logarithmic relationship, solving for a specific concentration \(t\) allows you to ensure the stability or reactivity of the solution remains within safe parameters. Understanding these algebraic manipulations ensures you can accurately predict and manage variables in scientific and financial models.

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