The image you provided contains a set of logarithmic problems written in Indonesian. Below is a breakdown of the questions and how to approach each of them:

1. Convert exponential expressions into logarithmic form

• a. $2^5$
  • Answer: This becomes $\log_2 32 = 5$ since $2^5 = 32$.
• b. $\left( \frac{1}{4} \right)^{-3}$
  • Answer: This becomes $\log_{\frac{1}{4}} 64 = -3$ because $\left( \frac{1}{4} \right)^{-3} = 64$.

2. Determine the value of the logarithmic form

• a. $\log_2 128$
  • Answer: $\log_2 128 = 7$ because $2^7 = 128$.
• b. $\log_3 27$
  • Answer: $\log_3 27 = 3$ because $3^3 = 27$.
• c. $\log_5 \frac{1}{125}$
  • Answer: $\log_5 \frac{1}{125} = -3$ because $5^{-3} = \frac{1}{125}$.
• d. $\log_7 \frac{1}{7}$
  • Answer: $\log_7 \frac{1}{7} = -1$ because $7^{-1} = \frac{1}{7}$.

3. Use logarithmic addition and subtraction properties

These problems require the use of the properties: $\log_b (xy) = \log_b x + \log_b y \quad \text{and} \quad \log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y$

• a. $\log 6 + \log 12 - \log 9$
  • This simplifies using properties: $\log \left( \frac{6 \times 12}{9} \right) = \log 8$.
• b. $\log 4 + \log 36 + \log 7 - \log 21$
  • This simplifies to: $\log \left( \frac{4 \times 36 \times 7}{21} \right) = \log 48$.
• c. $\frac{1}{2} \log 10 + \log 25 - \frac{1}{2} \log 100 - \log 20$
  • Simplified: $\frac{1}{2} \log 10 = \log \sqrt{10}, \log 25 = \log 5^2$, giving the result $\log 2.5$.

4. Use logarithmic properties to simplify

• Use properties like $a \log b = \log b^a$, $\log c \cdot \log d = \log c + \log d$, and other power and multiplication rules.
• a. $\log 5 + \log 16$
  • Simplifies to $\log 80$.
• b. $\log 6 + \log 8 - \log 9 - \log 2$
  • Simplifies to $\log \left( \frac{6 \times 8}{9 \times 2} \right) = \log \frac{48}{18} = \log \frac{8}{3}$.
• c. $\log 100 + \log_7 343 - \log 8$
  • Simplifies to $\log 1250$.
• d. $\log 27^{4/3} + \log_8^2 + \log_6 1 - \log \frac{1}{27}$
  • This simplifies to $\log 27 \times \log_8^2 = \log 81$.

5. Express logarithms in terms of $x$ and $y$

Assuming $\log 3 = x$ and $\log 5 = y$:

• a. $\log 10$
  • This can be expressed as $\log 10 = \log (2 \times 5) = \log 2 + \log 5$. Hence, $\log 10 = x + y$.
• b. $\log 25$
  • Express as $2 \log 5 = 2y$.
• c. $\frac{1}{5} \log 10$
  • Express as $\frac{1}{5} \times (x + y)$.
• d. $\log_{625} 200$
  • Express using $625 = 5^4$, simplifying into terms of $y$.

Let me know if you want detailed solutions to any specific question or further clarifications!

5 Related Questions:

1. How do we derive logarithmic properties from exponent rules?
2. What is the difference between natural logarithms and common logarithms?
3. Can you explain the change of base formula for logarithms?
4. How do logarithmic scales work in real-world applications, such as in sound intensity or earthquakes?
5. How do you calculate complex logarithmic expressions without a calculator?

Tip:

When simplifying logarithmic expressions, always look for factors that can combine terms using the basic properties of logarithms (addition for multiplication, subtraction for division, and exponents for powers).