Let's break down the solution step-by-step for the given equation:

$\log_3(3x + 4) - 7\log_9(x^2) + 6\log_3(x) = 2$

Step 1: Simplify $\log_9(x^2)$

We know that $\log_9(x^2)$ can be expressed in terms of $\log_3(x)$ because:

$\log_9(x^2) = \frac{1}{2} \log_3(x^2) = \frac{2}{2} \log_3(x) = \log_3(x)$

So the equation becomes:

$\log_3(3x + 4) - 7\log_3(x) + 6\log_3(x) = 2$

Step 2: Combine the logarithmic terms

We now combine the terms involving $\log_3(x)$:

$\log_3(3x + 4) - \log_3(x) = 2$

Step 3: Apply the logarithmic subtraction rule

Use the property of logarithms:

$\log_b(A) - \log_b(B) = \log_b\left(\frac{A}{B}\right)$

Thus, we apply this to the equation:

$\log_3\left(\frac{3x + 4}{x}\right) = 2$

Step 4: Convert the logarithmic equation to exponential form

Using the property of logarithms, $\log_b(A) = C$ implies $A = b^C$. Therefore:

$\frac{3x + 4}{x} = 3^2$

$\frac{3x + 4}{x} = 9$

Step 5: Solve the resulting equation

Now, solve the equation:

$3x + 4 = 9x$

$4 = 6x$

$x = \frac{4}{6} = \frac{2}{3}$

Step 6: Verify the solution

Substitute $x = \frac{2}{3}$ back into the original equation to verify the solution:

• $\log_3(3x + 4) = \log_3\left(3\left(\frac{2}{3}\right) + 4\right) = \log_3(2 + 4) = \log_3(6)$
• $\log_9(x^2) = \log_9\left(\left(\frac{2}{3}\right)^2\right) = \log_9\left(\frac{4}{9}\right)$, and convert this to base 3.
• Similarly check the remaining terms.

Thus, the solution $x = \frac{2}{3}$ is correct.

Let me know if you'd like to clarify any steps!

Related Questions:

1. How do you convert between logarithms of different bases?
2. What are the properties of logarithmic functions used in simplifying equations?
3. How can you solve logarithmic equations involving multiple variables?
4. What are the steps for verifying solutions in logarithmic equations?
5. How does the change of base formula work for logarithms?

Tip:

Always check your solutions by substituting them back into the original equation, especially in logarithmic equations where domain restrictions might occur!