The problem asks us to simplify the logarithmic expression: $log(\frac{15}{8}) + 4log2 - log3$.

1. Problem Description

The problem asks us to simplify the logarithmic expression:

$log(\frac{15}{8}) + 4log2 - log3$.

2. Solution Steps

We will use the properties of logarithms to simplify the expression.

First, we can rewrite $log(\frac{15}{8})$ using the quotient rule of logarithms:

$log(\frac{a}{b}) = log(a) - log(b)$

So, $log(\frac{15}{8}) = log(15) - log(8)$.

Now, we can rewrite $15$ as $3 \times 5$ and $8$ as $2^3$:

$log(15) = log(3 \times 5) = log(3) + log(5)$

$log(8) = log(2^3) = 3log(2)$

Therefore, $log(\frac{15}{8}) = log(3) + log(5) - 3log(2)$.

Next, consider the term $4log2$. Using the power rule of logarithms:

$alog(b) = log(b^a)$

$4log2 = log(2^4) = log(16)$.

Now, we can substitute these back into the original expression:

$log(\frac{15}{8}) + 4log2 - log3 = (log(3) + log(5) - 3log(2)) + log(16) - log(3)$

The $log(3)$ terms cancel out:

$log(5) - 3log(2) + log(16) = log(5) - log(2^3) + log(16) = log(5) - log(8) + log(16)$

Now, we can combine the logarithmic terms:

$log(5) + log(16) - log(8) = log(5 \times 16) - log(8) = log(80) - log(8)$

Using the quotient rule again:

$log(80) - log(8) = log(\frac{80}{8}) = log(10)$

Assuming the logarithm is base 10, $log_{10}(10) = 1$.

3. Final Answer

The final answer is

1.