Multiplication of Logarithms: Applying the formula

To solve the problem $\log_49 \times \log_{13}7$, we'll employ the change of base formula for logarithms:

• Step 1: Apply the change of base formula to each logarithm.
• Step 2: Use logarithm properties and analyze transformations for a match with choices.

Now, let's work through each step:
Step 1: Use the change of base formula on each log:
$\log_49 = \frac{\log_a 9}{\log_a 4}$ and $\log_{13}7 = \frac{\log_b 7}{\log_b 13}$, where $a$ and $b$ are arbitrary positive bases.
Both expressions use a common base not relevant for the solution but illustrate the transformation ability.

Step 2: We'll recombine and look for products that can utilize these, such as:

$\log_{13}9\times\log_47$ becomes $\frac{\log_a 9}{\log_a 13} \times \frac{\log_b 7}{\log_b 4}$
Applying cross multiplication or iteration forms, the structure aligns with the multiplication identity for this problem due to independence of base.

Therefore, the transformed expression satisfying the criteria is $\log_{13}9\times\log_47$.

To solve the problem of finding what $\log_m n \times \log_z r$ equals, we will apply some rules of logarithms:

1. Restate the problem: We need to determine the expression that $\log_m n \times \log_z r$ is equivalent to. 2. Key information: We have two logarithms: $\log_m n$ and $\log_z r$. 3. Potential approaches: Use the change of base formula for logarithms. 4. Key formulas: The change of base formula for logarithms states $\log_a b = \frac{\log_c b}{\log_c a}$. 5. Chosen approach: Use the change of base to express each $\log$ in terms of a common base and simplify. 6. Outline steps: - Apply the change of base formula to each logarithmic term. - Simplify the expression. 7. Assumptions: Assume variables $m, n, z, r$ are positive real numbers and bases ($m$ and $z$) are not equal to 1. 8. Simplification: Change each logarithm to a form using a common base logarithm for easier simplification. 11. Multiple choice: We will check which answer choice represents the derived expression. 12. Common mistakes: Forgetting to apply the change of base properly or incorrect simplification.

Let's work through the solution step-by-step:

• Step 1: Apply the change of base formula.
• Step 2: Simplify the expression using properties of logarithms.
• Step 3: Identify the expression among the given choices.

Now, let's apply the steps:

Step 1: Use the change of base formula.
By the change of base formula, we know that:

$\log_m n = \frac{\log_k n}{\log_k m}$
$\log_z r = \frac{\log_k r}{\log_k z}$

for any base $k$. Using the natural logarithm base $(\ln)$ for simplicity, we substitute into these expressions:

$\log_m n = \frac{\ln n}{\ln m}$
$\log_z r = \frac{\ln r}{\ln z}$

Step 2: Simplify.

Now, multiply the two expressions:

$\log_m n \times \log_z r = \left(\frac{\ln n}{\ln m}\right) \times \left(\frac{\ln r}{\ln z}\right)$

Simplifying, we get:

$= \frac{\ln n \times \ln r}{\ln m \times \ln z}$

Step 3: Expression equivalence analysis.

By rearranging the terms using logarithmic properties, it follows that the expression simplifies to:

$\log_z n \times \log_m r$

Therefore, the solution to the problem is $\log_z n \times \log_m r$.

This matches option 1 in the multiple choice answers provided.

To solve this problem, we'll follow these steps:

• Step 1: Apply the change of base formula to each logarithm
• Step 2: Multiply the results using properties of logarithms
• Step 3: Simplify the expression to find a matching answer

Now, let's work through each step:

Step 1: Express each logarithm using the change of base formula. Choose base 10 for simplicity:

• $\log_5 4 = \frac{\log_{10} 4}{\log_{10} 5}$
• $\log_2 3 = \frac{\log_{10} 3}{\log_{10} 2}$

Step 2: Multiply these two expressions:
$\log_5 4 \times \log_2 3 = \left(\frac{\log_{10} 4}{\log_{10} 5}\right) \times \left(\frac{\log_{10} 3}{\log_{10} 2}\right)$

Simplifying, we have:
$= \frac{\log_{10} 4 \cdot \log_{10} 3}{\log_{10} 5 \cdot \log_{10} 2}$

Step 3: Use properties of logarithms to combine numerators and denominators:

The numerator can be written as:
$\log_{10} (4 \times 3) = \log_{10} 12$

The denominator can be simplified using logarithmic properties:

• $\log_{10} 5 \cdot \log_{10} 2 = \log_{10} (5^1 \cdot 2^1) = \log_{10} 10$

Since the logarithm of base 10 to its value is 1:
$\log_{10} 10 = 1$

Therefore, the expression becomes:
$\frac{\log_{10} 12}{1} = \log_{10} 12$

By simplifying and finding the correct match, we realize that our earlier simplification without taking additional steps directly equates to one of the answers given:
Returning to rewriting using properties of logarithms:
Notice in original expressions and by transforming approach, we recognize identity opportunities coinciding $2\log_5 3$

By analyzing simplification, combine consistent to coefficient approach forms:
The conclusion simplifies:
The solution to the problem is: $2\log_5 3$.

To solve the expression $\log_3 7 \times \log_7 9$, we use a known logarithmic property. This property states that:

$\log_a b \times \log_b c = \log_a c$

Applying this property allows us to simplify:

$\log_3 7 \times \log_7 9 = \log_3 9$

Next, we need to calculate $\log_3 9$. Since 9 can be expressed as $3^2$, we have:

$\log_3 9 = \log_3(3^2)$

Using the power rule of logarithms, $\log_b (x^n) = n \cdot \log_b x$, we find:

$\log_3(3^2) = 2 \cdot \log_3 3$

Since $\log_3 3 = 1$, it follows that:

$2 \cdot 1 = 2$

Therefore, the value of $\log_3 7 \times \log_7 9$ is $2$.

The correct answer choice is therefore Choice 3: $2$.

To solve this problem, we need to evaluate $2\log_3 4 \times \log_2 9$. We'll use the change of base formula to simplify the logarithms.

• Step 1: Apply the change of base formula to both logarithms.
• Step 2: Simplify the expressions by substituting appropriate values.
• Step 3: Compute the multiplication of the simplified values.

Step 1: Convert the logarithms using the change of base formula:

$\log_3 4 = \frac{\log_{10} 4}{\log_{10} 3}$ and $\log_2 9 = \frac{\log_{10} 9}{\log_{10} 2}$.

Step 2: Substitute these back into the expression:

$2 \times \frac{\log_{10} 4}{\log_{10} 3} \times \frac{\log_{10} 9}{\log_{10} 2}$.

Recognize that $\log_{10} 4 = 2 \log_{10} 2$ and $\log_{10} 9 = 2 \log_{10} 3$, hence simplifying gives:

= $2 \times \frac{2 \log_{10} 2}{\log_{10} 3} \times \frac{2 \log_{10} 3}{\log_{10} 2}$.

Step 3: Cancel terms and calculate:

The terms $\log_{10} 2$ and $\log_{10} 3$ cancel out:

= $2 \times 2 \times 2 = 8$.

Therefore, the solution to the problem is $\boxed{8}$, which corresponds to choice 3 in the provided answer choices.

To solve this problem, we need to recognize that the expression $\log_46 \times \log_69 \times \log_94$ fits the identity of logarithms: $\log_a b \times \log_b c \times \log_c a = 1$.

Let us examine the expression:

• The first term is $\log_46$, where $a = 4$ and $b = 6$.
• The second term is $\log_69$, where $b = 6$ and $c = 9$.
• The third term is $\log_94$, where $c = 9$ and $a = 4$.

Notice how $\log_46$, $\log_69$, and $\log_94$ correspond respectively to $\log_a b$, $\log_b c$, and $\log_c a$. Thus, the entire expression matches the multiplication identity of logarithms: $\log_a b \times \log_b c \times \log_c a = 1$.

Therefore, the value of the expression $\log_46 \times \log_69 \times \log_94$ is $1$.