Let's go through the problems related to logarithms in the image step-by-step. The document is divided into three main sections. I'll address each question:

1. Express in logarithmic form:

A. $3^4 = 81$

• Logarithmic form: $\log_3 81 = 4$

B. $\sqrt{2} = 2^{\frac{1}{2}}$

• Logarithmic form: $\log_2 \sqrt{2} = \frac{1}{2}$

C. $0.001 = 10^{-3}$

• Logarithmic form: $\log_{10} 0.001 = -3$

2. Find the logarithmic values:

A. ${}^2\log 32$

• This means $\log_2 32$. Since $2^5 = 32$, we get: $\log_2 32 = 5$

B. ${}^3\log 3\sqrt{3}$

• This is $\log_3 (3\sqrt{3})$, which simplifies to: $\log_3 3 + \log_3 \sqrt{3} = 1 + \frac{1}{2} = 1.5$

C. ${}^2\log \frac{1}{2}\sqrt{2}$

• This becomes $\log_2 \left(\frac{1}{2}\sqrt{2}\right)$, which can be split as: $\log_2 \left( \frac{1}{2} \times 2^{1/2} \right) = \log_2 \left( 2^{-1} \times 2^{1/2} \right)$ Using the logarithmic property: $\log_2 (2^{-1}) + \log_2 (2^{1/2}) = -1 + \frac{1}{2} = -\frac{1}{2}$

3. Simplify:

A. ${}^2\log 4 + {}^2\log 8$

• This is $\log_2 4 + \log_2 8$, which becomes: $\log_2 (4 \times 8) = \log_2 32 = 5$

B. ${}^3\log \frac{1}{9} + {}^3\log 81$

• This becomes $\log_3 \frac{1}{9} + \log_3 81$, which simplifies to: $\log_3 \left( \frac{1}{9} \times 81 \right) = \log_3 9 = 2$

C. ${}^2\log 2\sqrt{2} + {}^2\log 4\sqrt{2}$

• Simplifies to: $\log_2 (2\sqrt{2}) + \log_2 (4\sqrt{2}) = \log_2 (2\sqrt{2} \times 4\sqrt{2}) = \log_2 32 = 5$

D. $\log 1,000 - \log 100$

• This is $\log 1,000 - \log 100 = \log \left( \frac{1,000}{100} \right) = \log 10 = 1$

E. ${}^3\log 18 - {}^3\log 6$

• This becomes $\log_3 \left( \frac{18}{6} \right) = \log_3 3 = 1$

F. ${}^3\log 7 \times {}^7\log 81$

• This involves multiple bases and requires further steps, but for now: ${}^3\log 7 \times {}^7\log 81 = \log_3 7 \times \log_7 81$ Use the change of base formula to simplify further.

G. $\text{If } {}^3\log 5 = a, \text{ calculate } 25 {}^{5}\log 27$

• This needs further calculation based on values of $a$ and logarithmic properties.

Would you like me to clarify any of these steps or continue with more details?

5 Related Questions:

1. How do you derive the change of base formula in logarithms?
2. What are the properties of logarithms useful for simplifying expressions?
3. Can you explain how to handle logarithms with fractional bases?
4. What are some practical applications of logarithms in real-world problems?
5. How do you calculate logarithmic values without a calculator?

Tip:

When simplifying logarithmic expressions, always check for common bases or opportunities to use the product, quotient, or power rules.