Question

solve for the exact value of x. \\(\log_{7}(5x) + 4\log_{7}(2) = 3\\)

Explanation:

Step1: Apply power rule to log term

$4\log_{7}(2) = \log_{7}(2^4) = \log_{7}(16)$

Step2: Combine log terms via product rule

$\log_{7}(5x) + \log_{7}(16) = \log_{7}(5x \times 16) = \log_{7}(80x)$

Step3: Convert log to exponential form

$7^3 = 80x$

Step4: Calculate $7^3$ and solve for x

$343 = 80x \implies x = \frac{343}{80}$

Answer:

$x = \frac{343}{80}$