Solve the following logarithmic equation. $$ \log _{3} x=3 $$

Question

UpStudy AI · Accepted Answer

To solve the logarithmic equation:

$$
\log_{3} x = 3
$$

**Step 1: Understand the Logarithmic Equation**

The equation $$\log_{3} x = 3$$ asks for the value of $$x$$ such that when you take the logarithm of $$x$$ with base $$3$$, you get $$3$$.

**Step 2: Rewrite the Logarithmic Equation in Exponential Form**

Recall that the logarithmic equation $$\log_{b} a = c$$ can be rewritten in its exponential form as:

$$
b^{c} = a
$$

Applying this to our equation:

$$
3^{3} = x
$$

**Step 3: Calculate the Exponential Expression**

Now, compute $$3^{3}$$:

$$
3^{3} = 3 \times 3 \times 3 = 27
$$

**Step 4: Write the Solution**

Thus, the solution to the equation is:

$$
x = 27
$$

**Verification (Optional):**

To verify, substitute $$x = 27$$ back into the original equation:

$$
\log_{3} 27 = 3
$$

Since $$3^{3} = 27$$, the equation holds true.

**Final Answer:**

$$
x = 27
$$
$$ x = 27 $$

Solve the following logarithmic equation.