The problem asks you to use the Power Rule for Logarithms to simplify or make an equivalent statement.

Power Rule for Logarithms:

$\log_b(x^n) = n \cdot \log_b(x)$

In this case: \log_7(1024) = \text{(rewrite using the power rule in terms of \log_7(4))}.

Step-by-Step Solution:

1. Recognize that $1024 = 4^5$.
2. Substitute $4^5$ into the logarithm: $\log_7(1024) = \log_7(4^5).$
3. Apply the power rule: $\log_7(4^5) = 5 \cdot \log_7(4).$

Final Answer:

$\log_7(1024) = 5 \cdot \log_7(4).$

Would you like further clarification or details?

Related Questions:

1. What is the base $b$ in logarithms and how does it affect calculations?
2. Can the Power Rule be applied to all logarithmic expressions?
3. How do logarithmic rules simplify complex expressions in practice?
4. How can you calculate $\log_7(4)$ if it’s not given explicitly?
5. Are there other rules, like the Product Rule or Quotient Rule, for logarithms?

Tip:

Always check if the argument in a logarithm can be expressed as a power of another number to apply the Power Rule effectively!