Question
Folve the following logarithmic possiale \( 6 \log _{2} x=-\log _{2} 64 \)
Ask by Bryant Coles. in the United States
Jan 22,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x = \frac{1}{2} \)
Solution
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:**
\[
x = \frac{1}{2}
\]
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Simplify this solution Extra Insights
To solve the equation \( 6 \log_{2} x = -\log_{2} 64 \), first, we can simplify the right side. The logarithm \(-\log_{2} 64\) can be rewritten since \(64 = 2^6\). Thus, \(-\log_{2} 64 = -6\). Now, substituting this back into the equation gives us: \( 6 \log_{2} x = -6 \). Next, divide both sides by 6: \( \log_{2} x = -1 \). To find \(x\), we convert the logarithmic form back to its exponential form, which tells us: \( x = 2^{-1} = \frac{1}{2} \). Therefore, the solution to the equation is \( x = \frac{1}{2} \).
