Solve the following logarithmic equation $$ \log _{6}(2 x)=3 $$

Question

UpStudy AI · Accepted Answer

To solve the logarithmic equation:

$$
\log_{6}(2x) = 3
$$

follow these steps:

### Step 1: Understand the Logarithmic Equation
The equation $$\log_{6}(2x) = 3$$ is in the form:

$$
\log_{b}(A) = C
$$

This can be rewritten in its exponential form as:

$$
A = b^{C}
$$

### Step 2: Rewrite the Equation in Exponential Form
Applying the above understanding to the given equation:

$$
\log_{6}(2x) = 3 \quad \Rightarrow \quad 2x = 6^{3}
$$

### Step 3: Calculate $$6^{3}$$
Compute the value of $$6^{3}$$:

$$
6^{3} = 6 \times 6 \times 6 = 216
$$

So, the equation becomes:

$$
2x = 216
$$

### Step 4: Solve for $$x$$
Divide both sides of the equation by 2 to isolate $$x$$:

$$
x = \frac{216}{2} = 108
$$

### Conclusion
The solution to the equation $$\log_{6}(2x) = 3$$ is:

$$
x = 108
$$
$$ x = 108 $$

Solve the following logarithmic equation \( \log _{6}(2 x)=3 \)