Question

what is the solution of \\(\log(2t+4) = \log(14-3t)\\)?

\\(\bigcirc\\) \\(-18\\)
\\(\bigcirc\\) \\(-2\\)
\\(\bigcirc\\) \\(2\\)
\\(\bigcirc\\) \\(10\\)

Explanation:

⚡ Using what you learned: Logarithms and Exponential Equations

Step 1: Equate the arguments

Since the bases of the logarithms on both sides are the same, their arguments must be equal:

$$ 2t + 4 = 14 - 3t $$

Step 2: Solve for \( t \)

Add \( 3t \) to both sides:

$$ 5t + 4 = 14 $$

Subtract \( 4 \) from both sides:

$$ 5t = 10 $$

Divide by \( 5 \):

$$ t = 2 $$

Step 3: Verify the solution

Check if \( t = 2 \) results in positive arguments for the logarithms:

$$ 2t + 4 = 2(2) + 4 = 8 > 0 $$

$$ 14 - 3t = 14 - 3(2) = 8 > 0 $$

Both arguments are positive, so the solution is valid.

Answer:

2