To solve the equation $3^{2x + 1} = 28$ using logarithms, follow these steps:

Step 1: Take the natural logarithm (ln) of both sides of the equation.

$\ln(3^{2x + 1}) = \ln(28)$

Step 2: Apply the logarithmic power rule.

The logarithmic power rule states that $\ln(a^b) = b \cdot \ln(a)$. Using this, we can move the exponent $2x + 1$ in front of the logarithm: $(2x + 1) \cdot \ln(3) = \ln(28)$

Step 3: Solve for $x$.

Now, distribute $\ln(3)$ on the left-hand side: $2x \cdot \ln(3) + 1 \cdot \ln(3) = \ln(28)$

Simplify further: $2x \cdot \ln(3) + \ln(3) = \ln(28)$

Now isolate $2x$: $2x \cdot \ln(3) = \ln(28) - \ln(3)$

Solve for $x$: $x = \frac{\ln(28) - \ln(3)}{2 \cdot \ln(3)}$

Step 4: Calculate the logarithms.

Using a calculator:

• $\ln(28) \approx 3.3322$
• $\ln(3) \approx 1.0986$

Now substitute these values into the equation: $x = \frac{3.3322 - 1.0986}{2 \cdot 1.0986}$ $x = \frac{2.2336}{2.1972}$ $x \approx 1.0166$

Final Answer:

$x \approx 1.0166$ (rounded to four decimal places).

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Would you like more details or have any questions?

Here are five related questions to explore:

1. How would you solve $5^{3x - 4} = 125$ using logarithms?
2. What is the logarithmic property that allows you to move exponents in front of the log?
3. Can you solve the equation $4^{x+2} = 50$ using logarithms?
4. How would you rewrite the equation $\log_b(x) = y$ in exponential form?
5. How can natural logarithms simplify equations involving powers?

Tip: Always use logarithms when trying to isolate variables in exponents!