Certainly! Let's find the exact value of the logarithmic expression [tex]\(\log_4 2 + \log_4 512\)[/tex] step by step.
### Step 1: Apply the Logarithmic Property
First, let's use the property of logarithms that states [tex]\(\log_b(x) + \log_b(y) = \log_b(x \cdot y)\)[/tex].
So, we can combine the given logarithms into a single logarithm:
[tex]\[
\log_4 2 + \log_4 512 = \log_4 (2 \cdot 512)
\][/tex]
### Step 2: Multiply the Arguments
Next, we need to calculate [tex]\(2 \cdot 512\)[/tex]:
[tex]\[
2 \cdot 512 = 1024
\][/tex]
Therefore,
[tex]\[
\log_4 (2 \cdot 512) = \log_4 1024
\][/tex]
### Step 3: Express the Argument as a Power of the Base
Now, we notice that 1024 can be expressed as a power of 4. Specifically, [tex]\(1024 = 4^5\)[/tex]. This is because when you repeatedly multiply 4 five times, you get:
[tex]\[
4^5 = 4 \times 4 \times 4 \times 4 \times 4 = 1024
\][/tex]
### Step 4: Simplify the Logarithm
By the properties of logarithms, [tex]\(\log_b(b^x) = x\)[/tex]. Hence,
[tex]\[
\log_4 1024 = \log_4 (4^5)
\][/tex]
[tex]\[
\log_4 (4^5) = 5
\][/tex]
### Final Answer
Thus, the exact value of the original logarithmic expression [tex]\(\log_4 2 + \log_4 512\)[/tex] is:
[tex]\[
5
\][/tex]
