https://file.aiquickdraw.com/mathbot/file/17293561267323bhkrecq.jpg

The image contains a set of logarithmic and exponential problems. I'll solve each of them step by step:

### 1. Problem:
\[
M = \log 625 - \log 1000 + \log 729
\]
\[
N = \log 8 - \log 81^2 + 49
\]
Find \( M + N \).

#### Solution for \( M \):
Using logarithmic properties:
- \( \log a - \log b = \log \left(\frac{a}{b}\right) \)
- \( \log a + \log b = \log(ab) \)

First simplify \( M \):
\[
M = \log \left(\frac{625 \times 729}{1000}\right)
\]
Now, calculate:
\[
625 = 5^4, \quad 1000 = 10^3, \quad 729 = 9^3 = (3^2)^3 = 3^6
\]
So:
\[
M = \log \left(\frac{5^4 \times 3^6}{10^3}\right)
\]
Combine:
\[
M = \log \left(\frac{(5^4)(3^6)}{10^3}\right)
\]
Now compute or simplify further later if necessary.

#### Solution for \( N \):
\[
N = \log 8 - \log 81^2 + 49
\]
Convert the expressions:
\[
8 = 2^3, \quad 81 = 9^2 = (3^2)^2 = 3^4
\]
Now simplify:
\[
N = \log(2^3) - \log(3^8) + 49
\]
Using logarithmic rules:
\[
N = 3\log 2 - 8\log 3 + 49
\]

At this point, you can evaluate both \( M \) and \( N \) numerically.

### 2. Problem:
\[
M = \log 16A, \quad \text{with } A = \log 5\pi
\]

### 3. Problem:
\[
\text{Find } \log_{75} 135
\]
This requires base conversion using the change of base formula:
\[
\log_b a = \frac{\log a}{\log b}
\]
So:
\[
\log_{75} 135 = \frac{\log 135}{\log 75}
\]

### 4. Problem:
\[
N = \log_2 \text{Antilog } 43/2
\]
This involves the concept of the antilogarithm and can be rewritten for simplification.

### 5. Problem:
\[
\text{If } \log 7 = 0.845, \text{ calculate } \log 70
\]
Using logarithmic properties:
\[
\log(70) = \log(7 \times 10) = \log 7 + \log 10 = 0.845 + 1 = 1.845
\]

### 6. Problem:
Solve:
\[
x^{\log x} = \left(\frac{10}{4}\right)^2
\]
Simplify:
\[
x^{\log x} = \left(\frac{5}{2}\right)^2 = \frac{25}{4}
\]

---

Would you like detailed solutions for any specific part of these problems? Here are five related questions:

1. How do you apply the change of base formula to solve logarithmic expressions?
2. What are the key properties of logarithms used to simplify expressions?
3. Can you simplify antilogarithmic expressions further?
4. How do you solve logarithmic equations involving powers?
5. What is the relation between logarithms and exponents in solving for unknown variables?

**Tip**: Always break down logarithmic problems using basic properties like addition, subtraction, and power rules for easier simplification.

Solve a series of logarithmic and exponential problems, such as calculating log differences, applying logarithmic properties, and solving logarithmic equations.

Solve complex logarithmic problems involving addition, subtraction, base conversion, and logarithmic equations. Learn step-by-step solutions using logarithmic properties and the change of base theorem. Suitable for high school students.

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