Certainly! To solve the logarithmic expression [tex]\(\log_8 5 + \log_8 7 = \log_8 \square\)[/tex], we can use the properties of logarithms. Here’s a detailed step-by-step explanation:
1. Recall the Property of Logarithms:
One of the key properties of logarithms is that the sum of two logarithms with the same base can be combined into a single logarithm. Specifically:
[tex]\[
\log_b(a) + \log_b(c) = \log_b(a \cdot c)
\][/tex]
This property is often referred to as the logarithm multiplication property.
2. Apply the Property:
Using the property above, we can combine the two logarithms in our expression:
[tex]\[
\log_8 5 + \log_8 7 = \log_8 (5 \cdot 7)
\][/tex]
3. Perform the Multiplication:
Next, we need to multiply the numbers inside the logarithm:
[tex]\[
5 \times 7 = 35
\][/tex]
4. Combine the Results:
So, the expression [tex]\(\log_8 (5 \cdot 7)\)[/tex] simplifies to:
[tex]\[
\log_8 (35)
\][/tex]
Therefore, the value inside the logarithm that completes the equation [tex]\(\log_8 5 + \log_8 7 = \log_8 \square\)[/tex] is [tex]\(35\)[/tex]. Hence, the completed equation is:
[tex]\[
\log_8 5 + \log_8 7 = \log_8 35
\][/tex]
So, [tex]\(\square = 35\)[/tex].
