 simplify the expression by using the definition and properties of logarithms. 2 tothepowerof log_2 5 - 3 log_5 ^3√5

To simplify the given expression:

\[
2^{\log_2 5} - 3 \log_5 \sqrt[3]{5}
\]

Let's break it down step by step.

### 1. Simplify \( 2^{\log_2 5} \):

This part involves a logarithm and an exponent with the same base. One of the properties of logarithms is:

\[
a^{\log_a x} = x
\]

Using this property for \( 2^{\log_2 5} \):

\[
2^{\log_2 5} = 5
\]

### 2. Simplify \( 3 \log_5 \sqrt[3]{5} \):

First, we rewrite \( \sqrt[3]{5} \) as an exponent:

\[
\sqrt[3]{5} = 5^{1/3}
\]

Next, apply the logarithmic property:

\[
\log_b(x^n) = n \log_b x
\]

This gives:

\[
\log_5 \sqrt[3]{5} = \log_5(5^{1/3}) = \frac{1}{3} \log_5 5
\]

Since \( \log_5 5 = 1 \), we have:

\[
\log_5 \sqrt[3]{5} = \frac{1}{3}
\]

Now, multiply by 3:

\[
3 \log_5 \sqrt[3]{5} = 3 \times \frac{1}{3} = 1
\]

### 3. Final Expression:

Now that both parts are simplified, we can subtract them:

\[
5 - 1 = 4
\]

Thus, the simplified expression is:

\[
\boxed{4}
\]

Would you like further details on this? 

Here are 5 related questions for practice:
1. Simplify \( 3^{\log_3 7} - 2 \log_7 \sqrt[4]{7} \).
2. What is \( 5^{\log_5 x} \) for any positive \( x \)?
3. Simplify \( 4 \log_2 \sqrt{8} \).
4. Use properties of logarithms to simplify \( \log_3 (9 \times 27) \).
5. Solve for \( x \): \( 2^{\log_2 (3x)} = 6 \).

**Tip:** Remember that logarithmic and exponential functions are inverses of each other!

Simplify the expression by using the definition and properties of logarithms: 2^log_2 5 - 3 log_5 (3√5)

This page provides a detailed step-by-step solution for simplifying the expression 2^log_2 5 - 3 log_5 (3√5) using properties of logarithms and exponential functions. The solution covers the logarithmic-exponential inverse theorem and key algebraic concepts. Ideal for students in Grades 9-11.

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