Simplify The Expression: $\log _4 16^3$A. $3 \log _4 16$ B. $\log _4 4096$ C. $\log _4 48$ D. $3 \log _4 4$

Simplify the Expression: $\log _4 16^3$

Understanding the Problem

The given problem involves simplifying an expression with logarithms. We are required to simplify the expression $\log _4 16^3$. To do this, we need to apply the properties of logarithms and simplify the expression step by step.

Applying the Power Rule of Logarithms

The power rule of logarithms states that $\log _a b^c = c \log _a b$. This rule allows us to simplify expressions with logarithms by bringing the exponent down as a coefficient.

Simplifying the Expression

Using the power rule of logarithms, we can simplify the expression $\log _4 16^3$ as follows:

$\log _4 16^3 = 3 \log _4 16$

This simplification is based on the power rule of logarithms, which states that $\log _a b^c = c \log _a b$. In this case, we have $\log _4 16^3$, which can be simplified to $3 \log _4 16$.

Evaluating the Simplified Expression

Now that we have simplified the expression to $3 \log _4 16$, we need to evaluate this expression. To do this, we need to find the value of $\log _4 16$.

Finding the Value of $\log _4 16$

To find the value of $\log _4 16$, we need to use the definition of logarithms. The logarithm $\log _a b$ is defined as the exponent to which we must raise the base $a$ to get the number $b$. In this case, we need to find the exponent to which we must raise the base $4$ to get the number $16$.

Using the Definition of Logarithms

Using the definition of logarithms, we can write $\log _4 16$ as an exponent:

$\log _4 16 = x$

where $x$ is the exponent to which we must raise the base $4$ to get the number $16$.

Solving for $x$

To solve for $x$, we can use the fact that $4^x = 16$. This is because the logarithm $\log _4 16$ is defined as the exponent to which we must raise the base $4$ to get the number $16$.

Finding the Value of $x$

To find the value of $x$, we can use the fact that $4^x = 16$. This can be rewritten as $4^x = 4^2$, since $16 = 4^2$.

Simplifying the Equation

Using the fact that $4^x = 4^2$, we can simplify the equation as follows:

$x = 2$

This means that the value of $\log _4 16$ is $2$.

Evaluating the Simplified Expression

Now that we have found the value of $\log _4 16$, we can evaluate the simplified expression $3 \log _4 16$.

Substituting the Value of $\log _4 16$

Substituting the value of $\log _4 16$ into the simplified expression, we get:

$3 \log _4 16 = 3 \cdot 2$

Simplifying the Expression

Simplifying the expression, we get:

$3 \log _4 16 = 6$

This means that the value of the expression $\log _4 16^3$ is $6$.

Conclusion

In conclusion, we have simplified the expression $\log _4 16^3$ using the power rule of logarithms. We have found that the value of the expression is $6$.

Answer

The correct answer is:

A. $3 \log _4 16$

However, this is not the final answer. We need to consider the other options as well.

Comparing the Options

Let's compare the options:

• Option A: $3 \log _4 16$
• Option B: $\log _4 4096$
• Option C: $\log _4 48$
• Option D: $3 \log _4 4$

Evaluating Option B

To evaluate option B, we need to find the value of $\log _4 4096$.

Finding the Value of $\log _4 4096$

Using the definition of logarithms, we can write $\log _4 4096$ as an exponent:

$\log _4 4096 = x$

where $x$ is the exponent to which we must raise the base $4$ to get the number $4096$.

Solving for $x$

To solve for $x$, we can use the fact that $4^x = 4096$. This is because the logarithm $\log _4 4096$ is defined as the exponent to which we must raise the base $4$ to get the number $4096$.

Finding the Value of $x$

To find the value of $x$, we can use the fact that $4^x = 4096$. This can be rewritten as $4^x = 4^6$, since $4096 = 4^6$.

Simplifying the Equation

Using the fact that $4^x = 4^6$, we can simplify the equation as follows:

$x = 6$

This means that the value of $\log _4 4096$ is $6$.

Evaluating Option C

To evaluate option C, we need to find the value of $\log _4 48$.

Finding the Value of $\log _4 48$

Using the definition of logarithms, we can write $\log _4 48$ as an exponent:

$\log _4 48 = x$

where $x$ is the exponent to which we must raise the base $4$ to get the number $48$.

Solving for $x$

To solve for $x$, we can use the fact that $4^x = 48$. This is because the logarithm $\log _4 48$ is defined as the exponent to which we must raise the base $4$ to get the number $48$.

Finding the Value of $x$

To find the value of $x$, we can use the fact that $4^x = 48$. This can be rewritten as $4^x = 4^3 \cdot 3$, since $48 = 4^3 \cdot 3$.

Simplifying the Equation

Using the fact that $4^x = 4^3 \cdot 3$, we can simplify the equation as follows:

$x = 3 + \log _4 3$

This means that the value of $\log _4 48$ is $3 + \log _4 3$.

Evaluating Option D

To evaluate option D, we need to find the value of $3 \log _4 4$.

Finding the Value of $3 \log _4 4$

Using the definition of logarithms, we can write $\log _4 4$ as an exponent:

$\log _4 4 = x$

where $x$ is the exponent to which we must raise the base $4$ to get the number $4$.

Solving for $x$

To solve for $x$, we can use the fact that $4^x = 4$. This is because the logarithm $\log _4 4$ is defined as the exponent to which we must raise the base $4$ to get the number $4$.

Finding the Value of $x$

To find the value of $x$, we can use the fact that $4^x = 4$. This can be rewritten as $4^x = 4^1$, since $4 = 4^1$.

Simplifying the Equation

Using the fact that $4^x = 4^1$, we can simplify the equation as follows:

$x = 1$

This means that the value of $\log _4 4$ is $1$.

Evaluating the Expression

Now that we have found the value of $\log _4 4$, we can evaluate the expression $3 \log _4 4$.

Substituting the Value of $\log _4 4$

Substituting the value of $\log _4 4$ into the expression, we get:

$3 \log _4 4 = 3 \cdot 1$

Simplifying the Expression

Simplifying the expression, we get:

$3 \log _4 4 = 3$

This means that the value of the expression $3 \log _4 4$ is $3$.

Conclusion

In conclusion, we have evaluated all the options and found that the correct answer is:

A. $3 \log _4 16$

However, we also found that the value of the expression $\log _4 16^3$ is $6$, which is not among the options. This means that the correct answer is not among the options.

Answer

The correct answer is:

A. $3 \log _4 16$

However, this is not the final answer. We need to consider the other options as well.

Final Answer

The final answer is:

A. $3 \log _4 16$
Simplify the Expression: $\log _4 16^3$ - Q&A

Q: What is the power rule of logarithms?

A: The power rule of logarithms states that $\log _a b^c = c \log _a b$. This rule allows us to simplify expressions with logarithms by bringing the exponent down as a coefficient.

Q: How do we simplify the expression $\log _4 16^3$ using the power rule of logarithms?

A: Using the power rule of logarithms, we can simplify the expression $\log _4 16^3$ as follows:

$\log _4 16^3 = 3 \log _4 16$

This simplification is based on the power rule of logarithms, which states that $\log _a b^c = c \log _a b$. In this case, we have $\log _4 16^3$, which can be simplified to $3 \log _4 16$.

Q: What is the value of $\log _4 16$?

A: To find the value of $\log _4 16$, we need to use the definition of logarithms. The logarithm $\log _a b$ is defined as the exponent to which we must raise the base $a$ to get the number $b$. In this case, we need to find the exponent to which we must raise the base $4$ to get the number $16$.

Q: How do we find the value of $\log _4 16$?

A: To find the value of $\log _4 16$, we can use the fact that $4^x = 16$. This is because the logarithm $\log _4 16$ is defined as the exponent to which we must raise the base $4$ to get the number $16$.

Q: What is the value of $x$ in the equation $4^x = 16$?

A: To find the value of $x$, we can use the fact that $4^x = 16$. This can be rewritten as $4^x = 4^2$, since $16 = 4^2$.

Q: How do we simplify the equation $4^x = 4^2$?

A: Using the fact that $4^x = 4^2$, we can simplify the equation as follows:

$x = 2$

This means that the value of $\log _4 16$ is $2$.

Q: What is the value of the expression $3 \log _4 16$?

A: Now that we have found the value of $\log _4 16$, we can evaluate the expression $3 \log _4 16$.

Q: How do we evaluate the expression $3 \log _4 16$?

A: Substituting the value of $\log _4 16$ into the expression, we get:

$3 \log _4 16 = 3 \cdot 2$

Q: What is the value of the expression $3 \log _4 16$?

A: Simplifying the expression, we get:

$3 \log _4 16 = 6$

This means that the value of the expression $\log _4 16^3$ is $6$.

Q: What is the correct answer among the options?

A: The correct answer is:

A. $3 \log _4 16$

However, we also found that the value of the expression $\log _4 16^3$ is $6$, which is not among the options. This means that the correct answer is not among the options.

Q: What is the final answer?

A: The final answer is:

A. $3 \log _4 16$

However, this is not the final answer. We need to consider the other options as well.

Q: What is the value of the expression $\log _4 4096$?

A: To find the value of $\log _4 4096$, we need to use the definition of logarithms. The logarithm $\log _a b$ is defined as the exponent to which we must raise the base $a$ to get the number $b$. In this case, we need to find the exponent to which we must raise the base $4$ to get the number $4096$.

Q: How do we find the value of $\log _4 4096$?

A: To find the value of $\log _4 4096$, we can use the fact that $4^x = 4096$. This is because the logarithm $\log _4 4096$ is defined as the exponent to which we must raise the base $4$ to get the number $4096$.

Q: What is the value of $x$ in the equation $4^x = 4096$?

A: To find the value of $x$, we can use the fact that $4^x = 4096$. This can be rewritten as $4^x = 4^6$, since $4096 = 4^6$.

Q: How do we simplify the equation $4^x = 4^6$?

A: Using the fact that $4^x = 4^6$, we can simplify the equation as follows:

$x = 6$

This means that the value of $\log _4 4096$ is $6$.

Q: What is the value of the expression $\log _4 48$?

A: To find the value of $\log _4 48$, we need to use the definition of logarithms. The logarithm $\log _a b$ is defined as the exponent to which we must raise the base $a$ to get the number $b$. In this case, we need to find the exponent to which we must raise the base $4$ to get the number $48$.

Q: How do we find the value of $\log _4 48$?

A: To find the value of $\log _4 48$, we can use the fact that $4^x = 48$. This is because the logarithm $\log _4 48$ is defined as the exponent to which we must raise the base $4$ to get the number $48$.

Q: What is the value of $x$ in the equation $4^x = 48$?

A: To find the value of $x$, we can use the fact that $4^x = 48$. This can be rewritten as $4^x = 4^3 \cdot 3$, since $48 = 4^3 \cdot 3$.

Q: How do we simplify the equation $4^x = 4^3 \cdot 3$?

A: Using the fact that $4^x = 4^3 \cdot 3$, we can simplify the equation as follows:

$x = 3 + \log _4 3$

This means that the value of $\log _4 48$ is $3 + \log _4 3$.

Q: What is the value of the expression $3 \log _4 4$?

A: To find the value of $3 \log _4 4$, we need to use the definition of logarithms. The logarithm $\log _a b$ is defined as the exponent to which we must raise the base $a$ to get the number $b$. In this case, we need to find the exponent to which we must raise the base $4$ to get the number $4$.

Q: How do we find the value of $\log _4 4$?

A: To find the value of $\log _4 4$, we can use the fact that $4^x = 4$. This is because the logarithm $\log _4 4$ is defined as the exponent to which we must raise the base $4$ to get the number $4$.

Q: What is the value of $x$ in the equation $4^x = 4$?

A: To find the value of $x$, we can use the fact that $4^x = 4$. This can be rewritten as $4^x = 4^1$, since $4 = 4^1$.

Q: How do we simplify the equation $4^x = 4^1$?

A: Using the fact that $4^x = 4^1$, we can simplify the equation as follows:

$x = 1$

This means that the value of $\log _4 4$ is $1$.

Q: What is the value of the expression $3 \log _4 4$?

A: Now that we have found the value of $\log _4 4$, we can evaluate the expression $3 \log _4 4$.

Q: How do we evaluate the expression $3 \log _4 4$?

A: Substituting the value of $\log _4 4$ into the expression, we get:

$3 \log _4 4 = 3 \cdot 1$

Q: What is the value of the expression $3 \log _4 4$?

A: Simplifying the expression, we get:

$3 \log _4 4 = 3$

This means that the value of the expression $3 \log _4 4$ is $3$.

Q: What is the final answer?

A: The final answer is:

A