To combine the expression [tex]\( 2 \log_4(9w + 1) + \frac{1}{2} \log_4(w + 8) \)[/tex] into a single logarithm, we can utilize the properties of logarithms. Here’s the step-by-step solution:
1. Use the Power Rule of Logarithms:
The power rule states that [tex]\( a \log_b(x) = \log_b(x^a) \)[/tex]. This allows us to bring coefficients from in front of the logarithms inside as exponents.
- For the first term: [tex]\( 2 \log_4(9w + 1) \)[/tex]:
[tex]\[
2 \log_4(9w + 1) = \log_4((9w + 1)^2)
\][/tex]
- For the second term: [tex]\( \frac{1}{2} \log_4(w + 8) \)[/tex]:
[tex]\[
\frac{1}{2} \log_4(w + 8) = \log_4((w + 8)^{1/2})
\][/tex]
Note that [tex]\( (w + 8)^{1/2} \)[/tex] is the same as [tex]\( \sqrt{w + 8} \)[/tex].
2. Sum of Logarithms:
The property that allows us to combine the sum of logarithms is [tex]\( \log_b(x) + \log_b(y) = \log_b(x \cdot y) \)[/tex].
- Combine the two logarithms:
[tex]\[
\log_4((9w + 1)^2) + \log_4((w + 8)^{1/2}) = \log_4((9w + 1)^2 \cdot (w + 8)^{1/2})
\][/tex]
3. Final Expression:
Combining the above steps, we get the single logarithm:
[tex]\[
\log_4((9w + 1)^2 \cdot (w + 8)^{1/2})
\][/tex]
So, the expression [tex]\( 2 \log_4(9w + 1) + \frac{1}{2} \log_4(w + 8) \)[/tex] as a single logarithm is:
[tex]\[
\log_4((9w + 1)^2 \cdot (w + 8)^{1/2})
\][/tex]
