Solve for x given log32𝑥+log36=log3 (8+4�

Let's solve the equation:

\[
\log_3(2x) + \log_3(6) = \log_3(8 + 4x)
\]

**Step 1: Apply the properties of logarithms**

Use the product property of logarithms, which states:

\[
\log_b(m) + \log_b(n) = \log_b(m \times n)
\]

Applying this to the left side:

\[
\log_3(2x \times 6) = \log_3(12x)
\]

So, the equation becomes:

\[
\log_3(12x) = \log_3(8 + 4x)
\]

**Step 2: Remove the logarithms**

Since the logarithms have the same base, we can equate the arguments:

\[
12x = 8 + 4x
\]

**Step 3: Solve for \(x\)**

Subtract \(4x\) from both sides:

\[
8x = 8
\]

Divide both sides by 8:

\[
x = 1
\]

So, the solution is \( x = 1 \).

Do you want details or have any questions? Here are 8 related questions:

1. What are the basic properties of logarithms?
2. How do you solve logarithmic equations with different bases?
3. What is the change of base formula for logarithms?
4. How can you use logarithms to solve exponential equations?
5. How do you simplify logarithmic expressions with coefficients?
6. What is the significance of the base in a logarithmic function?
7. How would the solution change if the equation had different coefficients?
8. Can logarithmic functions have negative arguments, and why?

**Tip:** When solving logarithmic equations, always check for extraneous solutions by substituting back into the original equation.

Learn how to solve the logarithmic equation log3(2x) + log3(6) = log3(8 + 4x) step-by-step with detailed explanations and examples. This problem involves logarithmic properties and equations, suitable for Grades 10-12.

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