Folve the following logarithmic possiale $$ 6 \log _{2} x=-\log _{2} 64 $$

Question

UpStudy AI · Accepted Answer

Sure, let's solve the logarithmic equation step by step:

$$
6 \log_{2} x = -\log_{2} 64
$$

### Step 1: Simplify the Right Side

First, simplify $$\log_{2} 64$$:

$$
\log_{2} 64 = \log_{2} (2^6) = 6
$$

So the equation becomes:

$$
6 \log_{2} x = -6
$$

### Step 2: Solve for $$\log_{2} x$$

Divide both sides of the equation by 6 to isolate $$\log_{2} x$$:

$$
\log_{2} x = \frac{-6}{6} = -1
$$

### Step 3: Exponentiate to Remove the Logarithm

To solve for $$x$$, rewrite the logarithmic equation in its exponential form. Recall that:

$$
\log_{b} a = c \quad \text{is equivalent to} \quad b^c = a
$$

Applying this to $$\log_{2} x = -1$$:

$$
2^{-1} = x
$$

$$
x = \frac{1}{2}
$$

### **Answer:**

$$
x = \frac{1}{2}
$$
$$ x = \frac{1}{2} $$

Folve the following logarithmic possiale \( 6 \log _{2} x=-\log _{2} 64 \)