https://file.aiquickdraw.com/mathbot/file/1726155275733ag8ihn5v.jpg

We are given that:

\[
\log_3 y = x \quad \text{and} \quad \log_2 z = x
\]

We need to find what \( 72^x \) is equal to in terms of \( y \) and \( z \).

### Step 1: Expressing \( y \) and \( z \) in terms of \( x \)
From \( \log_3 y = x \), we can express \( y \) as:
\[
y = 3^x
\]

From \( \log_2 z = x \), we can express \( z \) as:
\[
z = 2^x
\]

### Step 2: Breaking down \( 72^x \)
We know that:
\[
72 = 2^3 \times 3^2
\]

So, \( 72^x \) can be written as:
\[
72^x = (2^3 \times 3^2)^x = 2^{3x} \times 3^{2x}
\]

### Step 3: Substituting \( y = 3^x \) and \( z = 2^x \)
We can now substitute \( y = 3^x \) and \( z = 2^x \) into the expression:
\[
72^x = 2^{3x} \times 3^{2x}
\]

Notice that \( 2^{3x} = (2^x)^3 = z^3 \) and \( 3^{2x} = (3^x)^2 = y^2 \). Therefore, we have:
\[
72^x = z^3 \times y^2
\]

Thus, \( 72^x = y^2 z^3 \).

### Answer:
The correct option is **(c)**: \( y^2 z^3 \).

---

Would you like more details or have any other questions? Here are 5 related questions to expand on this topic:

1. How would the solution change if the logarithmic bases were different?
2. Can you prove why \( \log_b a = x \) implies \( a = b^x \)?
3. What is the general form of simplifying powers of products like \( (a \cdot b)^n \)?
4. How would the result change if the exponent was \( x^2 \) instead of \( x \)?
5. How does the relationship between logarithms and exponents help in simplifying expressions?

**Tip:** Remember that logarithmic identities can often be used to convert between multiplication and exponentiation, making it easier to simplify complex expressions!

If log₃ y = x and log₂ z = x, then 72ˣ is equal to:

Solve the problem involving logarithms and exponents: If log₃ y = x and log₂ z = x, find the value of 72ˣ in terms of y and z. This solution uses properties of logarithms and exponents to simplify the expression. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation