Let's solve each problem step by step using the
product rule and
quotient rule of logarithms:
---
🔷 Logarithmic Rules Used:
1.
Product Rule:
$$
\log_a M + \log_a N = \log_a (M \cdot N)
$$
2.
Quotient Rule:
$$
\log_a M - \log_a N = \log_a \left(\frac{M}{N}\right)
$$
3.
Power Rule (if needed):
$$
\log_a M^n = n \log_a M
$$
We'll apply these to rewrite each expression as a
single logarithm.
---
✔ Problem 1:
$$
\log_a x + \log_a y - \log_a z
$$
Step 1: Use product rule on the first two terms:
$$
\log_a x + \log_a y = \log_a (xy)
$$
Step 2: Now subtract the third term using quotient rule:
$$
\log_a (xy) - \log_a z = \log_a \left(\frac{xy}{z}\right)
$$
✔ Answer:
$$
\boxed{\log_a \left(\frac{xy}{z}\right)}
$$
---
✔ Problem 2:
$$
\log_s l^4 m^2 + \log_s n - \log_s lm
$$
Step 1: Combine the first two logs using product rule:
$$
\log_s (l^4 m^2) + \log_s n = \log_s (l^4 m^2 \cdot n) = \log_s (l^4 m^2 n)
$$
Step 2: Subtract the last log:
$$
\log_s (l^4 m^2 n) - \log_s (lm) = \log_s \left(\frac{l^4 m^2 n}{lm}\right)
$$
Step 3: Simplify the fraction:
- $ l^4 / l = l^3 $
- $ m^2 / m = m $
- So, numerator becomes: $ l^3 m n $
$$
= \log_s (l^3 m n)
$$
✔ Answer:
$$
\boxed{\log_s (l^3 m n)}
$$
---
✔ Problem 3:
$$
\log_h \frac{x}{y} + \log_h \frac{y}{z} - \log_h \frac{z}{y}
$$
Step 1: Combine the first two logs using product rule:
$$
\log_h \left(\frac{x}{y} \cdot \frac{y}{z}\right) = \log_h \left(\frac{x}{z}\right)
$$
Step 2: Now subtract the third log:
$$
\log_h \left(\frac{x}{z}\right) - \log_h \left(\frac{z}{y}\right) = \log_h \left(\frac{x/z}{z/y}\right)
$$
Step 3: Simplify the complex fraction:
$$
\frac{x/z}{z/y} = \frac{x}{z} \cdot \frac{y}{z} = \frac{xy}{z^2}
$$
✔ Answer:
$$
\boxed{\log_h \left(\frac{xy}{z^2}\right)}
$$
---
✔ Problem 4:
$$
\log_d ab + \log_d bc - \log_d ca
$$
Step 1: Combine the first two logs:
$$
\log_d (ab) + \log_d (bc) = \log_d (ab \cdot bc) = \log_d (a b^2 c)
$$
Step 2: Subtract the third log:
$$
\log_d (a b^2 c) - \log_d (ca) = \log_d \left(\frac{a b^2 c}{c a}\right)
$$
Step 3: Simplify:
- $ a/a = 1 $
- $ c/c = 1 $
- So: $ \frac{a b^2 c}{c a} = b^2 $
✔ Answer:
$$
\boxed{\log_d (b^2)}
$$
> Alternatively, you could write it as $ 2 \log_d b $, but since the question asks for a
single logarithm, $ \log_d (b^2) $ is acceptable.
---
✔ Problem 5:
$$
\log_s (m+n) + \log_s (l+m) - \log_s m
$$
Step 1: Combine the first two logs:
$$
\log_s [(m+n)(l+m)]
$$
Step 2: Subtract the third log:
$$
\log_s \left(\frac{(m+n)(l+m)}{m}\right)
$$
✔ Answer:
$$
\boxed{\log_s \left(\frac{(m+n)(l+m)}{m}\right)}
$$
---
✔ Final Answers Summary:
| Problem | Answer |
|--------|--------|
| 1 | $\log_a \left(\frac{xy}{z}\right)$ |
| 2 | $\log_s (l^3 m n)$ |
| 3 | $\log_h \left(\frac{xy}{z^2}\right)$ |
| 4 | $\log_d (b^2)$ |
| 5 | $\log_s \left(\frac{(m+n)(l+m)}{m}\right)$ |
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Let me know if you'd like this formatted as a printable worksheet or need explanations in simpler terms!
Parent Tip: Review the logic above to help your child master the concept of printable math worksheets power rule.
