Change-of-Base Formula for Logarithms: Applying the formula

To solve this problem, let's simplify the given expression $\frac{\log_85}{\log_89}$.

• Step 1: Recognize that both the numerator and denominator have the same base, 8.
• Step 2: The division property of logarithms states that $\frac{\log_b M}{\log_b N} = \log_N M$.
• Step 3: Apply the division rule to the given expression: $\frac{\log_8 5}{\log_8 9} = \log_9 5$.

Thus, after simplifying, we see that $\frac{\log_85}{\log_89} = \log_9 5$.

Hence, the correct answer is $\log_9 5$, which corresponds to the choice 1.

To solve the problem of evaluating $\log_7 4$, we will use the change-of-base formula for logarithms.

The change-of-base formula is:

• $\log_b a = \frac{\log_k a}{\log_k b}$, where $k$ can be any base, commonly chosen as 10 (common logs) or $e$ (natural logs).

We will choose natural logarithms ($\ln$) for simplicity, therefore:

$\log_7 4 = \frac{\ln 4}{\ln 7}$

By applying the change-of-base formula, we find that the logarithm $\log_7 4$ can be expressed as $\frac{\ln 4}{\ln 7}$.

Upon examining the provided choices, we identify that choice 2: $\frac{\ln 4}{\ln 7}$ matches our result.

Therefore, the solution to the problem is $\frac{\ln 4}{\ln 7}$.

To solve the given expression $\frac{\log_{4x}9}{\log_{4x}a}$ using the change-of-base formula, follow these steps:

• Step 1: Apply the change-of-base formula to both the numerator and the denominator expressions.
  This gives us: $\log_{4x}9 = \frac{\log_a 9}{\log_a (4x)}$ and $\log_{4x}a = \frac{\log_a a}{\log_a (4x)}$.
• Step 2: Substitute these into our original expression:
  $\frac{\log_{4x}9}{\log_{4x}a} = \frac{\frac{\log_a 9}{\log_a (4x)}}{\frac{\log_a a}{\log_a (4x)}}$.
• Step 3: Simplify the fraction:
  The $\log_a (4x)$ cancels out from the numerator and the denominator, leaving us with $\frac{\log_a 9}{\log_a a}$.
• Step 4: Further simplify using the fact that $\log_a a = 1$ because any number $a$ to the power of 1 is $a$.
  This results in $\frac{\log_a 9}{1} = \log_a 9$.

Therefore, the expression simplifies to $\log_a 9$.

The correct answer is $\log_a 9$, which matches choice 1.

To solve this problem, we'll simplify the given expression $\frac{\log_9e^2}{\log_9e}$ using logarithmic rules:

Step 1: Apply the power rule of logarithms:
The numerator $\log_9e^2$ can be rewritten using the power rule: $\log_9e^2 = 2 \cdot \log_9e$.

Step 2: Substitute and simplify the fraction:
Substitute the expression from Step 1 into the original problem:
$\frac{\log_9e^2}{\log_9e} = \frac{2 \cdot \log_9e}{\log_9e}$.

Step 3: Cancel common terms:
Since $\log_9e$ appears in both the numerator and the denominator, it cancels out, leaving:
$\frac{2 \cdot \log_9e}{\log_9e} = 2$.

Therefore, the solution to the problem is $\boxed{2}$.

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

• Step 1: Apply the mathematical property using the quotient rule for logarithms.
• Step 2: Simplify the expression into the desired format.

Now, let's work through each step:
Step 1: Given the expression $\frac{\log_8 9a}{\log_8 3a}$, we can directly apply the quotient rule for logarithms, which tells us that $\frac{\log_b M}{\log_b N} = \log_N M$.
Step 2: Applying this formula, we find that $\frac{\log_8 9a}{\log_8 3a} = \log_{3a} 9a$.

Therefore, the solution to the problem is $\log_{3a} 9a$.

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

• Step 1: Identify the expression $\ln 4x$.
• Step 2: Apply the change-of-base formula to transform the natural logarithm to one using base 7.
• Step 3: Use the formula $\ln a = \frac{\log_b a}{\log_b e}$ to substitute the values.

Now, let's work through each step:
Step 1: The expression given is $\ln 4x$.
Step 2: We want a base 7 logarithm, so we apply the change-of-base formula:
Step 3: We have:

$\ln 4x = \frac{\log_7 4x}{\log_7 e}$

Therefore, the logarithmic expression $\ln 4x$ in base 7 is equivalent to $\frac{\log_7 4x}{\log_7 e}$.

This matches the correct answer choice, which is choice 4.