5) $$ \quad 2: 3^{2 x+1}=510 $$

Question

UpStudy AI · Accepted Answer

To solve the exponential equation:

$$
3^{2x + 1} = 510
$$

**Step 1: Take the Natural Logarithm of Both Sides**

To solve for $$ x $$, we'll use logarithms. Taking the natural logarithm (ln) of both sides gives:

$$
\ln(3^{2x + 1}) = \ln(510)
$$

**Step 2: Apply the Power Rule of Logarithms**

The power rule states that $$ \ln(a^b) = b \ln(a) $$. Applying this:

$$
(2x + 1) \ln(3) = \ln(510)
$$

**Step 3: Solve for $$ x $$**

First, isolate $$ 2x + 1 $$:

$$
2x + 1 = \frac{\ln(510)}{\ln(3)}
$$

Then, solve for $$ x $$:

$$
2x = \frac{\ln(510)}{\ln(3)} - 1
$$

$$
x = \frac{1}{2} \left( \frac{\ln(510)}{\ln(3)} - 1 \right)
$$

**Step 4: Calculate the Numerical Value**

Let's compute the values using a calculator:

1. Calculate $$ \ln(510) $$:
   $$
   \ln(510) \approx 6.2313
   $$
2. Calculate $$ \ln(3) $$:
   $$
   \ln(3) \approx 1.0986
   $$
3. Compute the ratio:
   $$
   \frac{\ln(510)}{\ln(3)} \approx \frac{6.2313}{1.0986} \approx 5.67
   $$
4. Substitute back into the equation for $$ x $$:
   $$
   x = \frac{1}{2} (5.67 - 1) = \frac{1}{2} \times 4.67 \approx 2.335
   $$

**Final Answer:**

$$
x \approx 2.34
$$
$$ x \approx 2.34 $$

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