Sure, let's solve the equation [tex]\(\log_3 x - \log_3 2 = 1\)[/tex] step-by-step.
### Step 1: Use the properties of logarithms
One of the properties of logarithms states that the difference of two logarithms with the same base can be combined into a single logarithm by division:
[tex]\[
\log_b a - \log_b c = \log_b \left(\frac{a}{c}\right)
\][/tex]
Applying this property to the given equation:
[tex]\[
\log_3 x - \log_3 2 = \log_3 \left(\frac{x}{2}\right)
\][/tex]
So, our equation now becomes:
[tex]\[
\log_3 \left(\frac{x}{2}\right) = 1
\][/tex]
### Step 2: Rewrite the logarithmic equation as an exponential equation
To solve for [tex]\(x\)[/tex], we need to rewrite the logarithmic equation in its exponential form. Recall that:
[tex]\[
\log_b a = c \quad \text{is equivalent to} \quad a = b^c
\][/tex]
Using this property, we get:
[tex]\[
\frac{x}{2} = 3^1
\][/tex]
### Step 3: Simplify the exponential equation
Calculating the right-hand side:
[tex]\[
3^1 = 3
\][/tex]
So, the equation becomes:
[tex]\[
\frac{x}{2} = 3
\][/tex]
### Step 4: Solve for [tex]\(x\)[/tex]
To isolate [tex]\(x\)[/tex], multiply both sides of the equation by 2:
[tex]\[
x = 3 \times 2
\][/tex]
Thus:
[tex]\[
x = 6
\][/tex]
### Conclusion
The solution to the equation [tex]\(\log_3 x - \log_3 2 = 1\)[/tex] is [tex]\(x = 6\)[/tex].
