Question

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

Explanation:

Step1: Apply log addition rule

Using the property $\log_a(M) + \log_a(N) = \log_a(MN)$, we combine the logarithms:
$\log_3(2x \cdot 4) = 5$
Simplify the argument:
$\log_3(8x) = 5$

Step2: Convert to exponential form

Recall that $\log_a(b) = c$ is equivalent to $a^c = b$. So we have:
$3^5 = 8x$

Step3: Calculate $3^5$ and solve for x

$3^5 = 243$, so:
$243 = 8x$
Divide both sides by 8:
$x = \frac{243}{8}$

Answer:

$\frac{243}{8}$