Question

what is the solution of \\(\log(7t - 2) = \log(17t - 42)\\)?

\\(\bigcirc\\) 4
\\(\bigcirc\\) 8
\\(\bigcirc\\) 18
\\(\bigcirc\\) 20

Explanation:

⚡ Using what you learned: Logarithms and Exponential Equations

Step 1: Equate the arguments

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

$$ 7x - 8 = 17x - 48 $$

Step 2: Solve for x

Rearrange the terms to isolate \( x \):

$$ -8 + 48 = 17x - 7x $$

$$ 40 = 10x $$

$$ x = 4 $$

Step 3: Verify the solution

Check if \( x = 4 \) makes the arguments of the logarithms positive:

• For \( 7x - 8 \):

$$ 7(4) - 8 = 28 - 8 = 20 > 0 $$

• For \( 17x - 48 \):

$$ 17(4) - 48 = 68 - 48 = 20 > 0 $$

Both arguments are positive, so the solution is valid.

Answer:

4