Questions: Solve the following logarithmic equation. log5(t+16) - log5(t+7) = log5 t Select the correct choice below and, if necessary, fill in the answer box. A. The solution(s) is/are . (Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as needed.) B. The solution is not a real number.

Solve the following logarithmic equation.
log5(t+16) - log5(t+7) = log5 t

Select the correct choice below and, if necessary, fill in the answer box.
A. The solution(s) is/are  .
(Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as needed.)
B. The solution is not a real number.
Transcript text: Solve the following logarithmic equation. \[ \log _{5}(t+16)-\log _{5}(t+7)=\log _{5} t \] Select the correct choice below and, if necessary, fill in the answer box. A. The solution(s) is/are $\square$ $\square$. (Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as needed.) B. The solution is not a real number.
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Solution

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Solve the logarithmic equation \( \log_{5}(t+16) - \log_{5}(t+7) = \log_{5} t \).

Combine the logarithmic expressions.

Using the property of logarithms, we rewrite the left side as \( \log_{5}\left(\frac{t+16}{t+7}\right) = \log_{5} t \).

Set the arguments equal to each other.

This gives us the equation \( \frac{t+16}{t+7} = t \).

Cross-multiply to eliminate the fraction.

We obtain \( t + 16 = t(t + 7) \), which simplifies to \( t^2 + 7t - t - 16 = 0 \) or \( t^2 + 6t - 16 = 0 \).

Solve the quadratic equation.

Factoring gives us \( (t - 2)(t + 8) = 0 \), leading to solutions \( t = 2 \) and \( t = -8 \).

Check the validity of the solutions.

Only \( t = 2 \) is valid since logarithms are defined for positive arguments.

The solution is \( \boxed{2} \).

The solution is \( \boxed{2} \).

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