Sure, let's work through the problem step-by-step to write [tex]\( 2 \log_b p + \frac{1}{2} \log_b q - 5 \log_b r \)[/tex] as a single logarithm.
1. Apply the power rule of logarithms:
The power rule states that [tex]\( a \log_b x = \log_b (x^a) \)[/tex]. We apply this rule to each term in the expression:
- For [tex]\( 2 \log_b p \)[/tex], we use the power rule to get [tex]\( \log_b (p^2) \)[/tex].
- For [tex]\( \frac{1}{2} \log_b q \)[/tex], we use the power rule to get [tex]\( \log_b (q^{\frac{1}{2}}) \)[/tex]. Note that [tex]\( q^{\frac{1}{2}} \)[/tex] is the same as [tex]\( \sqrt{q} \)[/tex].
- For [tex]\( -5 \log_b r \)[/tex], we use the power rule to get [tex]\( \log_b (r^{-5}) \)[/tex].
2. Rewrite the expression with the modified logarithms:
Substituting the above results yields:
[tex]\[
\log_b (p^2) + \log_b (q^{\frac{1}{2}}) - \log_b (r^5)
\][/tex]
3. Combine the logarithms using the addition and subtraction rules:
- The addition rule of logarithms says [tex]\( \log_b x + \log_b y = \log_b (xy) \)[/tex].
- The subtraction rule of logarithms says [tex]\( \log_b x - \log_b y = \log_b \left(\frac{x}{y}\right) \)[/tex].
So, we first combine the addition part:
[tex]\[
\log_b (p^2) + \log_b (q^{\frac{1}{2}}) = \log_b (p^2 \cdot q^{\frac{1}{2}})
\][/tex]
Then, incorporate the subtraction part:
[tex]\[
\log_b (p^2 \cdot q^{\frac{1}{2}}) - \log_b (r^5) = \log_b \left( \frac{p^2 \cdot q^{\frac{1}{2}}}{r^5} \right)
\][/tex]
4. Simplify the expression:
- [tex]\( q^{\frac{1}{2}} \)[/tex] is the same as [tex]\( \sqrt{q} \)[/tex].
Thus, the final combined single logarithm expression is:
[tex]\[
\log_b \left(\frac{p^2 \sqrt{q}}{r^5}\right)
\][/tex]
Therefore, the expression [tex]\( 2 \log_b p + \frac{1}{2} \log_b q - 5 \log_b r \)[/tex] written as a single logarithm is:
[tex]\[
\log_b \left( \frac{p^2 \sqrt{q}}{r^5} \right)
\][/tex]
