To simplify the given logarithmic expression [tex]\(5 \log_2 k - 8 \log_2 m + 10 \log_2 n\)[/tex], we will use the properties of logarithms.
1. Product Property: [tex]\(\log_b(x) + \log_b(y) = \log_b(xy)\)[/tex]
2. Quotient Property: [tex]\(\log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right)\)[/tex]
3. Power Property: [tex]\(a \log_b(x) = \log_b(x^a)\)[/tex]
Using these properties, we can simplify the expression step-by-step:
### Step 1: Apply the Power Property
Rewrite each term by applying the Power Property:
[tex]\[
5 \log_2 k = \log_2 (k^5)
\][/tex]
[tex]\[
8 \log_2 m = \log_2 (m^8)
\][/tex]
[tex]\[
10 \log_2 n = \log_2 (n^{10})
\][/tex]
### Step 2: Substitute the rewritten terms back into the expression
The expression now is:
[tex]\[
\log_2 (k^5) - \log_2 (m^8) + \log_2 (n^{10})
\][/tex]
### Step 3: Apply the Quotient and Product Properties
First, combine the logs using the Quotient Property:
[tex]\[
\log_2 (k^5) - \log_2 (m^8) = \log_2 \left(\frac{k^5}{m^8}\right)
\][/tex]
Next, use the Product Property to include the [tex]\(\log_2 (n^{10})\)[/tex]:
[tex]\[
\log_2 \left(\frac{k^5}{m^8}\right) + \log_2 (n^{10}) = \log_2 \left(\frac{k^5 \cdot n^{10}}{m^8}\right)
\][/tex]
### Final simplified expression
So, the simplified form of the expression [tex]\(5 \log_2 k - 8 \log_2 m + 10 \log_2 n\)[/tex] is:
[tex]\[
\log_2 \left(\frac{k^5 n^{10}}{m^8}\right)
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{\log_2 \frac{k^5 n^{10}}{m^8}}
\][/tex]
