find exact value of expression. 4(log47+log47)

The given expression is:

\[
4 \left( \log_4 7 + \log_4 7 \right)
\]

First, simplify the terms inside the parentheses:

\[
\log_4 7 + \log_4 7 = 2 \log_4 7
\]

Now, substitute this into the original expression:

\[
4 \times 2 \log_4 7 = 8 \log_4 7
\]

Next, let's rewrite the expression using a logarithmic identity. Recall that \( a \log_b x = \log_b (x^a) \). Applying this identity, we get:

\[
8 \log_4 7 = \log_4 7^8 = \log_4 (7^8)
\]

Now, calculate \( 7^8 \):

\[
7^8 = 5764801
\]

Thus, the expression becomes:

\[
\log_4 5764801
\]

This is the exact value of the expression.

Would you like more details or clarifications?

Here are 5 related questions to explore:
1. What are the properties of logarithms used to simplify this expression?
2. How do you convert logarithms with different bases?
3. Can logarithms be used to solve exponential equations?
4. How does the change of base formula for logarithms work?
5. What are the applications of logarithmic identities in real-world problems?

**Tip:** Always simplify inside parentheses first when working with logarithmic expressions.

find exact value of expression. 4(log4 7 + log4 7)

Learn how to find the exact value of the logarithmic expression 4(log4 7 + log4 7) using logarithmic identities and exponents. Step-by-step explanation provided, suitable for students in Grades 9-12.

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