I have one way, but cant think of another?
Show two ways of finding y if y = log4 64. Explain your thinking.
4^x=64
x=3
64 = 4^3
so log_4(64) = 3log_4(4) = 3*1 = 3
To find the value of y when y = log4 64, there are two ways you can approach it:
Method 1: Using Exponentiation
1. Recall that in logarithmic form, loga b = x means that ax = b.
2. In this case, y = log4 64 translates to 4^y = 64.
3. Notice that 64 is equal to 4^3, so plug in 3 for y in the equation 4^y = 64.
4. Therefore, y = 3.
Method 2: Using the Change of Base Formula
1. The logarithm function with base 4 can be expressed using the change of base formula, which states that loga b = logc b / logc a.
2. In this case, y = log4 64 can be rewritten as y = log(64) / log(4).
3. Evaluate log(64) and log(4) using any base you prefer (commonly used bases are 10 and e).
- For example, using base 10: log(64) ≈ 1.806 and log(4) = 0.602.
4. Divide log(64) by log(4): 1.806 / 0.602 ≈ 3.
5. Therefore, y ≈ 3.
Both methods lead to the same conclusion that y = 3.
To find the value of y if y = log4 64, we can use two different methods.
Method 1: Change of Base Formula
Step 1: Apply the change of base formula to rewrite the logarithm in terms of a more familiar base, such as log base 10 or log base e.
log4 64 = log(64) / log(4)
Step 2: Evaluate the logarithms on the right side using a calculator or a logarithm table.
log(64) ≈ 1.806
log(4) ≈ 0.602
Step 3: Divide the values obtained in step 2 to find y.
log(64) / log(4) ≈ 1.806 / 0.602 ≈ 3
Therefore, y ≈ 3.
Method 2: Using the Relationship between Logs and Exponents
Step 1: Express the logarithmic equation in exponential form.
y = log4 64 can be rewritten as 4^y = 64.
Step 2: Write the base and the result in terms of the same base.
If we write 64 as 4^3, the equation becomes 4^y = 4^3.
Step 3: Since the bases are equal, we can equate the exponents.
This gives us y = 3.
Therefore, y = 3.
Both methods yield the same result: y = 3.