If `5.21^(p) = 2.86^(q)` , what is the value of `p/q` ?

Options

A. `-0.60`
B. 0.55
C. 0.6
D. 0.64

Explanation

To solve the equation \(5.21^p = 2.86^q\) and find the value of \(\frac{p}{q}\), we can use logarithms. Here’s a step-by-step solution:

### Step 1: Take the logarithm of both sides
We start by taking the logarithm of both sides of the equation:
\[ \log(5.21^p) = \log(2.86^q) \]

**Hint:** Remember that taking the logarithm allows us to bring the exponent down as a coefficient.

### Step 2: Apply the logarithmic property
Using the property of logarithms that states \(\log(b^a) = a \cdot \log(b)\), we can rewrite both sides:
\[ p \cdot \log(5.21) = q \cdot \log(2.86) \]

**Hint:** This property is useful for simplifying equations involving exponents.

### Step 3: Isolate \(\frac{p}{q}\)
To find \(\frac{p}{q}\), we can rearrange the equation:
\[ \frac{p}{q} = \frac{\log(2.86)}{\log(5.21)} \]

**Hint:** Dividing both sides by \(q\) helps isolate the ratio we are looking for.

### Step 4: Calculate the logarithms
Next, we need to calculate the logarithms:
- \(\log(5.21) \approx 0.719\)
- \(\log(2.86) \approx 0.456\)

**Hint:** Use a scientific calculator or logarithm tables to find these values accurately.

### Step 5: Substitute the logarithm values
Now, substitute these values into the equation:
\[ \frac{p}{q} = \frac{0.456}{0.719} \]

**Hint:** This step involves simple division.

### Step 6: Perform the division
Now, calculate the division:
\[ \frac{p}{q} \approx 0.634 \]

**Hint:** Ensure you perform the division carefully to get an accurate result.

### Final Answer
Thus, the value of \(\frac{p}{q}\) is approximately \(0.634\).