Question

1. solve for \\(x\\) \\(\log 2^{2x} = \log 3^{5-x}\\)

Explanation:

Apply power property of logarithms

We start with the given equation:

$$ \log 2^{2x} = \log 3^{5-x} $$

Using the power property of logarithms, we bring the exponents to the front:

$$ 2x \log 2 = (5 - x) \log 3 $$

Expand the equation

We distribute \(\log 3\) on the right side:

$$ 2x \log 2 = 5 \log 3 - x \log 3 $$

Group terms with x

We add \(x \log 3\) to both sides to collect all terms containing \(x\) on the left:
Using the Linear Equations concept:

$$ 2x \log 2 + x \log 3 = 5 \log 3 $$

Factor out x

We factor out \(x\) from the left side of the equation:

$$ x (2 \log 2 + \log 3) = 5 \log 3 $$

Solve for x

We divide both sides by \((2 \log 2 + \log 3)\) to isolate \(x\):

$$ x = \frac{5 \log 3}{2 \log 2 + \log 3} $$

Using logarithm properties, we can simplify the denominator:

$$ 2 \log 2 = \log 2^2 = \log 4 $$

$$ 2 \log 2 + \log 3 = \log 4 + \log 3 = \log 12 $$

Thus, the exact solution is:

$$ x = \frac{5 \log 3}{\log 12} $$

Answer:

$$x = \frac{5 \log 3}{\log 12}$$