Fill in the missing values to make the equations true.=+(a)log45log49log4=−(b)log8log87log837=(c)log99log93

To solve for the missing values in the equations provided, we will break down each part systematically.

1. ### Break Down the Problem

We need to fill in the missing values indicated by the symbols $+$ and $-$ to make the logarithmic equations true. We'll handle each equation individually.

Equation (a): [

• (a) \log_4 5 \log_4 9 \log_4 = ? ]

Equation (b): [

• (b) \log_8 \log_8 7 \log_8 37 = ? ]

Equation (c): $(c) \log_{99} \log_{93} = ?$

2. ### Relevant Concepts

1. Logarithmic Identities:

   • $\log_b (xy) = \log_b x + \log_b y$
   • $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$
   • $\log_b (x^y) = y \log_b x$

2. Property of Logarithms for Simplification:

   • Recognizing logarithms can be translated into exponential form.

3. ### Analysis and Detail

For (a): We need to find the relationship between $\log_4 5$, $\log_4 9$, and the value for $(a)$. Using properties, $\log_4 9 = \log_4 (3^2) = 2 \log_4 3$ Thus, we can express $\log_4 5 + 2 \log_4 3 = ?$ To make this equation true, we can find an approximation or representation of $(a)$.

For (b): $(a) \text{ should negate the product of the logs. Hence, it becomes:}$ $\log_8 7 - \log_8 37 = ?$ Using properties, we can simplify this further.

For (c): $(c) \log_{99} \log_{93}$ Indicates a relationship in terms of both logarithms, potentially indicating correlations related to the base and the logs.

4. ### Verify and Summarize

Checking against logarithmic properties for common bases or values will yield the desired outputs for $a$, $b$, and $c$.

Final Answer

Without specific numeric values provided within the equations, they cannot be calculated directly. However, having guided through the steps, we infer the relationships based on the properties of logarithms. The answer for $a$, $b$, and $c$ can't be conclusively derived without specific logarithmic inputs; however:

• $a$ relates to accessing $\log_4 5 + 2 \log_4 3$.
• $b$ connects with $\log_8 7 - \log_8 37$.
• $c$ states $\log_{99} \log_{93}$.

This breakdown illustrates how logarithmic relationships can define each missing variable.