How do you solve log4y 9 log44 log464?

To solve the expression log₄y 9 log₄4 log₄64, we need to break it down step by step. Let’s start simplifying each logarithm:

1. **Calculating log₄4**:

We know that log_bb = 1 for any base b (in this case, base 4). So, log₄4 = 1.

2. **Calculating log₄64**:

Since 64 is equal to 4 raised to the power of 3 (because 4 = 2² and 64 = 2⁶, hence 64 = (2²)³), we have:

log₄64 = log₄(4³) = 3.

3. **Substituting the values**:

Now we can substitute the calculated values back into the original equation:

log4y 9 log44 log464 = log4y 9 * 1 * 3

4. **Simplifying further**:

This simplifies to:

3 * log4y 9

5. **Understanding log₄y 9**:

Next, log₄y 9 means the logarithm of 9 to the base 4. A common way to express this in terms of natural logarithm (or log base 10) is by the change of base formula:

log4y 9 = logy9 / logy4

Hence, our final expression becomes:

3 * (logy9 / logy4)

To summarize, the solution to log₄y 9 log₄4 log₄64 ultimately simplifies to:

3 * (logy9 / logy4)