Sure, let's simplify the expression [tex]\( 5 \log_2 k - 8 \log_2 m + 10 \log_2 n \)[/tex] step-by-step.
Given:
[tex]\[ 5 \log_2 k - 8 \log_2 m + 10 \log_2 n \][/tex]
We can use the properties of logarithms to simplify this.
### Step 1: Factor out the logarithms
One important property of logarithms is that [tex]\( a \log_b x = \log_b (x^a) \)[/tex]. Using this, we can rewrite each term:
[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: Combine the logarithms
We can now combine these logarithms using another property: [tex]\( \log_b a + \log_b b = \log_b (a \cdot b) \)[/tex] and [tex]\( \log_b a - \log_b b = \log_b \left(\frac{a}{b}\right) \)[/tex].
Combining the terms:
[tex]\[ \log_2 (k^5) + \log_2 (n^{10}) - \log_2 (m^{-8}) \][/tex]
### Step 3: Further simplification
Now, using the property of logarithms that says the sum of the logs is the log of the product and the difference of logs is the log of the quotient, we get:
[tex]\[ \log_2 \left(\frac{k^5 n^{10}}{m^8}\right) \][/tex]
Therefore, the simplified form of [tex]\( 5 \log_2 k - 8 \log_2 m + 10 \log_2 n \)[/tex] is:
[tex]\[ \boxed{\log_2 \left(\frac{k^5 n^{10}}{m^8}\right)} \][/tex]
So, the correct simplified form is:
[tex]\[ \log_2 \left(\frac{k^5 n^{10}}{m^8}\right) \][/tex]
