Let's go through each of these logarithmic equations step by step and fill in the missing values.

(a) $\log_2 9 - \log_2 5 = \log_2 (\_\_\_\_)$

We can simplify the left-hand side using the logarithmic identity:

$\log_b x - \log_b y = \log_b \left(\frac{x}{y}\right)$

Thus,

$\log_2 9 - \log_2 5 = \log_2 \left(\frac{9}{5}\right)$

So, the missing value is $\frac{9}{5}$.

(b) $\log_3 8 + \log_3 (\_\_\_\_) = \log_3 88$

Here we can use the logarithmic property:

$\log_b x + \log_b y = \log_b (x \cdot y)$

Therefore,

$\log_3 8 + \log_3 (\_\_\_\_) = \log_3 (8 \times \_\_\_\_) = \log_3 88$

This implies:

$8 \times (\_\_\_\_) = 88$

Solving for the missing value:

$(\_\_\_\_) = \frac{88}{8} = 11$

Thus, the missing value is 11.

(c) $-2 \log_8 2 = \log_8 (\_\_\_\_)$

We use the property of logarithms where a coefficient in front of the logarithm can be rewritten as an exponent:

$a \log_b x = \log_b x^a$

So,

$-2 \log_8 2 = \log_8 (2^{-2}) = \log_8 \left(\frac{1}{4}\right)$

Thus, the missing value is $\frac{1}{4}$.

Summary of Missing Values:

• (a) $\frac{9}{5}$
• (b) 11
• (c) $\frac{1}{4}$

Let me know if you want further details on any step or have any questions.

5 Related Questions:

1. What are the properties of logarithms used in these equations?
2. How do you solve logarithmic equations involving subtraction or addition?
3. Can you explain how to convert between logarithmic and exponential forms?
4. How do the coefficients in front of a logarithm affect its value?
5. What other logarithmic identities are commonly used in simplifying expressions?

Tip:

When solving logarithmic equations, always look for opportunities to apply properties like the product, quotient, and power rules of logarithms.