The image contains five logarithmic expressions, and I'll help solve them step by step. Let's start by simplifying each of the logarithmic equations one by one.

1. $^2\log 18 + ^2\log 6 - ^2\log 27$

This can be simplified using the logarithmic properties:

• $^2\log a + ^2\log b = ^2\log (a \cdot b)$
• $^2\log a - ^2\log b = ^2\log \left(\frac{a}{b}\right)$

Applying this: $^2\log (18 \cdot 6) - ^2\log 27 = ^2\log \left(\frac{18 \cdot 6}{27}\right)$ $^2\log \left(\frac{108}{27}\right) = ^2\log 4$

So, the expression simplifies to $^2\log 4 = 2$.

2. $^3\log 72 - 2 \cdot ^3\log 4 + ^3\log 2$

Again using the properties:

• $n \cdot ^b\log a = ^b\log (a^n)$
• $^b\log a - ^b\log b = ^b\log \left(\frac{a}{b}\right)$

Simplifying step by step: $^3\log 72 - 2 \cdot ^3\log 4 + ^3\log 2 = ^3\log 72 - ^3\log (4^2) + ^3\log 2$ $= ^3\log 72 - ^3\log 16 + ^3\log 2 = ^3\log \left(\frac{72 \cdot 2}{16}\right)$ $= ^3\log \left(\frac{144}{16}\right) = ^3\log 9$

So, the expression simplifies to $^3\log 9 = 2$.

3. $\log 300 + 2 \cdot \log 3 - \log 0.27$

Simplifying the terms: $2 \cdot \log 3 = \log 3^2 = \log 9$ Using the property: $\log a + \log b = \log (a \cdot b), \quad \log a - \log b = \log \left(\frac{a}{b}\right)$ $\log 300 + \log 9 - \log 0.27 = \log \left(\frac{300 \cdot 9}{0.27}\right)$ $= \log \left(\frac{2700}{0.27}\right) = \log 10000$

Thus, the result is $\log 10000 = 4$.

4. $^2\log 16 - ^3\log \frac{1}{27} + ^5\log 1$

Simplifying:

• $^b\log 1 = 0$ (logarithm of 1 is always 0)
• $^b\log a^{-1} = - ^b\log a$ So: $^2\log 16 - ^3\log \frac{1}{27} + ^5\log 1 = ^2\log 16 + ^3\log 27 + 0$ Since $^2\log 16 = 4$ and $^3\log 27 = 3$, the expression becomes: $4 + 3 = 7$

5. $\log 24 - 2 \cdot \log 2 + \log \frac{1}{6}$

Simplifying: $2 \cdot \log 2 = \log 2^2 = \log 4$ Using logarithmic properties: $\log 24 - \log 4 + \log \frac{1}{6} = \log \left(\frac{24}{4}\right) + \log \frac{1}{6}$ $= \log 6 + \log \frac{1}{6} = \log \left(6 \cdot \frac{1}{6}\right)$ $= \log 1 = 0$

Thus, the expression simplifies to 0.

Summary of Results:

1. $2$
2. $2$
3. $4$
4. $7$
5. $0$

Would you like more details on any of these solutions?

Related Questions:

1. What are the basic properties of logarithms?
2. How do logarithmic functions relate to exponential functions?
3. What are common mistakes made when simplifying logarithmic expressions?
4. How do you apply logarithmic transformations in real-world problems?
5. What are the different bases commonly used in logarithms (e.g., natural log, base-10 log)?

Tip:

Remember that the logarithmic identities $\log a + \log b = \log(ab)$ and $\log a - \log b = \log \left(\frac{a}{b}\right)$ are your best tools for simplifying expressions.