Power Property of Logarithms Practice Problems & Solutions

To solve this problem, let's simplify $2\log_3 8$ using logarithm rules.

• Step 1: Recognize the expression form
  The expression is of the form $a \cdot \log_b c$, where $a = 2$, $b = 3$, and $c = 8$.
• Step 2: Apply the power property
  According to the power property of logarithms, $2 \cdot \log_3 8$ can be simplified to $\log_3 (8^2)$.
• Perform the calculation
  Calculate $8^2$, which is $64$.
• Step 3: Simplify further
  Therefore, we have $\log_3 64$.

This is a straightforward application of the power property of logarithms. By applying this property correctly, we've simplified the original expression correctly.

Therefore, the simplified form of $2\log_3 8$ is $\log_3 64$.

To simplify the expression $3\log_76$, we apply the power property of logarithms, which states:

$a\log_b c = \log_b(c^a)$

Step 1: Identify the given expression: $3\log_76$.

Step 2: Apply the power property of logarithms:

$3\log_76 = \log_7(6^3)$

Step 3: Calculate $6^3$:

$6^3 = 6 \times 6 \times 6 = 36 \times 6 = 216$

Step 4: Substitute back into the logarithmic expression:

$\log_7(6^3) = \log_7216$

Therefore, the simplified expression is $\log_7216$.

Comparing with the answer choices, the correct choice is:

$\log_7216$

To solve this problem, we'll follow the steps outlined:

• Step 1: Recognize that the expression $x \ln 7$ can be thought of in terms of the power property of logarithms, which helps reframe it into a single logarithm.
• Step 2: Apply the formula $\ln(a^b) = b \ln a$. This tells us that if we have something of the form $b \ln a$, we can express it as $\ln(a^b)$.
• Step 3: Utilize the known expression and rule by substituting $a = 7$ and $b = x$. Thus, $x \ln 7$ becomes $\ln(7^x)$.

Therefore, the rewritten expression for $x \ln 7$ using logarithm rules is $\ln 7^x$.

This matches choice 4 from the provided options.

To solve the problem $\log_6 8$, we need to express the number 8 as a power of a base that simplifies the logarithm. We can write 8 as $2^3$, because 8 equals 2 multiplied by itself three times.

Let's use the power property of logarithms, which is:

• $\log_b (a^n) = n \log_b a$

Applying this property to $\log_6 8$, we have:

$\log_6 8 = \log_6 (2^3)$

Using the power property, this becomes:

$\log_6 (2^3) = 3 \log_6 2$

Therefore, the expression for $\log_6 8$ in terms of $\log_6 2$ is:

$3 \log_6 2$.

To solve this problem, we need to transform the expression $n\log_xa$ using the properties of logarithms.

• Step 1: Identify the expression: We are given $n\log_xa$, where $\log_xa$ is the logarithm of $a$ to the base $x$, and $n$ is a coefficient.
• Step 2: Use the power property of logarithms: The power property of logarithms states that if we have a logarithmic term multiplied by a coefficient $n$, like $n\log_b(a)$, it can be rewritten as $\log_b(a^n)$.
• Step 3: Apply the power property: By applying this property to $n\log_xa$, we rewrite it as $\log_x(a^n)$. This is because multiplying the logarithmic term by an external coefficient is equivalent to taking the argument $a$ to the power of that coefficient, $n$.
• Step 4: Conclusion about the transformation: This transformation demonstrates how the power property helps simplify expressions involving logarithms by turning multiplication into an exponentiation within the logarithm itself.

Therefore, the expression $n\log_xa$ can be transformed and expressed as $\log_xa^n$ by using the power property of logarithms.