Question: logb M = logb M - logb N
loge M = ln M ln - logb? me Natural
R esule
log
log > 3 + 4\cdot\sqrt{y}
log > 3 + logz \cdot 1 + logz \sqrt{y}
log_{s} 3 + 4/logz \cdot x + 1/2 logc \sqrt
1. **Stating the problem:**
The user provides several logarithmic expressions and seems to be exploring logarithm properties and inequalities involving logarithms.
2. **Recall logarithm properties:**
- $\log_b M - \log_b N = \log_b \left(\frac{M}{N}\right)$
- $\log_e M = \ln M$ (natural logarithm)
- $\log_b (MN) = \log_b M + \log_b N$
- $\log_b (M^k) = k \log_b M$
3. **Analyzing the inequality:**
Given: $\log > 3 + 4 \cdot \sqrt{y}$
This expression is incomplete but suggests an inequality involving logarithms and a term $4 \cdot \sqrt{y}$.
4. **Rewrite the second expression:**
$\log > 3 + \log_z \cdot 1 + \log_z \sqrt{y}$
Using logarithm properties:
- $\log_z \cdot 1 = 0$ since $\log_z 1 = 0$
- $\log_z \sqrt{y} = \log_z y^{1/2} = \frac{1}{2} \log_z y$
So the right side simplifies to $3 + 0 + \frac{1}{2} \log_z y$
5. **Third expression:**
$\log_s 3 + \frac{4}{\log_z} \cdot x + \frac{1}{2} \log_c \sqrt{}$
This expression is incomplete and ambiguous, but we can note:
- $\log_s 3$ is the logarithm base $s$ of 3
- $\frac{4}{\log_z} \cdot x$ suggests multiplication of $x$ by $\frac{4}{\log_z}$
- $\frac{1}{2} \log_c \sqrt{}$ is incomplete but likely means $\frac{1}{2} \log_c (\text{something})$
6. **Summary:**
The user is working with logarithm properties, including subtraction, multiplication, and natural logarithms.
7. **Final note:**
Without a specific question or complete expressions, the key takeaway is to apply logarithm rules carefully:
$$\log_b M - \log_b N = \log_b \left(\frac{M}{N}\right)$$
$$\log_b (MN) = \log_b M + \log_b N$$
$$\log_b (M^k) = k \log_b M$$
$$\log_e M = \ln M$$
No specific numeric answer can be given due to incomplete expressions.