Evaluate The Logarithmic Expression.$\log _8 8^f$Select The Correct Choice Below And, If Necessary, Fill In The Answer Box To Complete Your Choice.A. $\log _8 8^f =$ $\square$B. The Logarithm Is Undefined.

Understanding Logarithmic Expressions

When dealing with logarithmic expressions, it's essential to understand the properties and rules that govern them. A logarithmic expression is a mathematical operation that represents the power or exponent to which a base number must be raised to produce a given value. In this case, we're given the expression $\log _8 8^f$, where the base is 8 and the argument is $8^f$.

The Logarithm of a Power Property

One of the fundamental properties of logarithms is the logarithm of a power property, which states that $\log _b a^c = c \log _b a$. This property allows us to simplify complex logarithmic expressions by bringing the exponent down as a coefficient.

Applying the Logarithm of a Power Property

To evaluate the expression $\log _8 8^f$, we can apply the logarithm of a power property. By substituting $a = 8$ and $c = f$ into the property, we get:

$$\log _8 8^f = f \log _8 8$$

Evaluating the Logarithm of the Base

Now, we need to evaluate the logarithm of the base, which is $\log _8 8$. Since the base and the argument are the same, we can use the property that $\log _b b = 1$. Therefore, $\log _8 8 = 1$.

Substituting the Value of the Logarithm of the Base

Substituting the value of the logarithm of the base into the expression, we get:

$$\log _8 8^f = f \cdot 1 = f$$

Conclusion

Based on the logarithm of a power property and the property that $\log _b b = 1$, we can conclude that $\log _8 8^f = f$. Therefore, the correct choice is:

A. $\log _8 8^f =$ $\boxed{f}$

Alternative Choice: The Logarithm is Undefined

If the base and the argument of a logarithmic expression are not the same, the logarithm is undefined. However, in this case, the base and the argument are the same, so the logarithm is defined.

Final Answer

The final answer is $\boxed{f}$.

Frequently Asked Questions

In this section, we'll address some common questions related to evaluating logarithmic expressions, specifically the expression $\log _8 8^f$.

Q: What is the logarithm of a power property?

A: The logarithm of a power property states that $\log _b a^c = c \log _b a$. This property allows us to simplify complex logarithmic expressions by bringing the exponent down as a coefficient.

Q: How do I apply the logarithm of a power property to the expression $\log _8 8^f$?

A: To apply the logarithm of a power property, substitute $a = 8$ and $c = f$ into the property. This gives us $\log _8 8^f = f \log _8 8$.

Q: What is the value of $\log _8 8$?

A: Since the base and the argument are the same, we can use the property that $\log _b b = 1$. Therefore, $\log _8 8 = 1$.

Q: How do I simplify the expression $\log _8 8^f$ using the value of $\log _8 8$?

A: Substitute the value of $\log _8 8$ into the expression, giving us $\log _8 8^f = f \cdot 1 = f$.

Q: Is the logarithm $\log _8 8^f$ defined?

A: Yes, the logarithm $\log _8 8^f$ is defined because the base and the argument are the same.

Q: What is the final answer to the expression $\log _8 8^f$?

A: The final answer is $\boxed{f}$.

Q: Can I use the logarithm of a power property to simplify any logarithmic expression?

A: Yes, the logarithm of a power property can be used to simplify any logarithmic expression of the form $\log _b a^c$, where $a$ and $b$ are positive real numbers and $c$ is a real number.

Q: What are some common mistakes to avoid when evaluating logarithmic expressions?

A: Some common mistakes to avoid when evaluating logarithmic expressions include:

• Forgetting to apply the logarithm of a power property when the exponent is not 1.
• Not using the property that $\log _b b = 1$ when the base and the argument are the same.
• Not simplifying the expression correctly after applying the logarithm of a power property.

Additional Resources

For more information on logarithmic expressions and their properties, see the following resources:

• Logarithmic Expression Properties
• Logarithmic Expression Simplification
• Logarithmic Expression Evaluation

Conclusion

Evaluating logarithmic expressions can be a complex task, but by understanding the properties and rules that govern them, you can simplify even the most complex expressions. Remember to apply the logarithm of a power property, use the property that $\log _b b = 1$, and simplify the expression correctly to get the final answer.