Simplify: `log((3)/(8)) + log ((45)/(8)) - log ((15)/(16))`.

Answer

To simplify the expression `log((3)/(8)) + log((45)/(8)) - log((15)/(16))`, we can follow these steps:

### Step 1: Combine the first two logarithms
Using the property of logarithms that states `log(a) + log(b) = log(a * b)`, we can combine the first two terms:
\[ log\left(\frac{3}{8}\right) + log\left(\frac{45}{8}\right) = log\left(\frac{3}{8} \cdot \frac{45}{8}\right) \]
Calculating the product:
\[ \frac{3 \cdot 45}{8 \cdot 8} = \frac{135}{64} \]
Thus, we have:
\[ log\left(\frac{135}{64}\right) \]

### Step 2: Subtract the third logarithm
Now, we will subtract the third logarithm using the property `log(a) - log(b) = log(a / b)`:
\[ log\left(\frac{135}{64}\right) - log\left(\frac{15}{16}\right) = log\left(\frac{\frac{135}{64}}{\frac{15}{16}}\right) \]
This simplifies to:
\[ log\left(\frac{135}{64} \cdot \frac{16}{15}\right) \]

### Step 3: Simplify the fraction
Now we simplify the expression inside the logarithm:
\[ \frac{135 \cdot 16}{64 \cdot 15} \]
Calculating the numerator and denominator:
- The numerator: \(135 \cdot 16 = 2160\)
- The denominator: \(64 \cdot 15 = 960\)

Thus, we have:
\[ log\left(\frac{2160}{960}\right) \]

### Step 4: Simplify the fraction further
Now we can simplify \(\frac{2160}{960}\):
\[ \frac{2160 \div 960}{960 \div 960} = \frac{2160 \div 960}{1} = \frac{27}{12} = \frac{9}{4} \]

### Step 5: Final expression
Now we have:
\[ log\left(\frac{9}{4}\right) \]

### Step 6: Rewrite in terms of simpler logs
We can express \(\frac{9}{4}\) as:
\[ log\left(\left(\frac{3}{2}\right)^2\right) = 2 \cdot log\left(\frac{3}{2}\right) \]

### Final Answer
Thus, the simplified expression is:
\[ 2 \cdot log\left(\frac{3}{2}\right) \]