Given that `log_2a=lamda,log_4b=lamda^2` and `log_(c^2)(8)=2/(lamda^3+1)` write `log_2((a^2b^5)/5)` as a function of `lamda,(a,b,cgt0,cne1)`.

Answer

To solve the problem step by step, we will use the properties of logarithms and the given equations.

### Step 1: Write the expression in logarithmic form
We need to find \( \log_2 \left( \frac{a^2 b^5}{5} \right) \). We can use the properties of logarithms to separate this expression:

\[ \log_2 \left( \frac{a^2 b^5}{5} \right) = \log_2 (a^2 b^5) - \log_2 (5) \]

### Step 2: Expand the logarithm of the product
Using the property \( \log_b (xy) = \log_b x + \log_b y \), we can expand \( \log_2 (a^2 b^5) \):

\[ \log_2 (a^2 b^5) = \log_2 (a^2) + \log_2 (b^5) \]

Using the property \( \log_b (x^n) = n \log_b x \):

\[ \log_2 (a^2) = 2 \log_2 a \quad \text{and} \quad \log_2 (b^5) = 5 \log_2 b \]

Thus,

\[ \log_2 (a^2 b^5) = 2 \log_2 a + 5 \log_2 b \]

### Step 3: Substitute the values of \( \log_2 a \) and \( \log_2 b \)
From the problem, we are given:

1. \( \log_2 a = \lambda \)
2. \( \log_4 b = \lambda^2 \)

We can convert \( \log_4 b \) to base 2:

\[ \log_4 b = \frac{\log_2 b}{\log_2 4} = \frac{\log_2 b}{2} \]

Thus,

\[ \log_2 b = 2 \lambda^2 \]

Now substituting these values into our expression:

\[ \log_2 (a^2 b^5) = 2 \lambda + 5 (2 \lambda^2) = 2 \lambda + 10 \lambda^2 \]

### Step 4: Combine everything together
Now we can substitute back into our original expression:

\[ \log_2 \left( \frac{a^2 b^5}{5} \right) = (2 \lambda + 10 \lambda^2) - \log_2 (5) \]

### Final Expression
Thus, the final expression for \( \log_2 \left( \frac{a^2 b^5}{5} \right) \) as a function of \( \lambda \) is:

\[ \log_2 \left( \frac{a^2 b^5}{5} \right) = 10 \lambda^2 + 2 \lambda - \log_2 (5) \]