Let's solve the given logarithmic equation step by step:
[tex]\[ 2 \log x - \log \frac{x}{2} = -1 \][/tex]
1. Simplify the logarithmic terms:
Recall the properties of logarithms:
- [tex]\( \log \frac{a}{b} = \log a - \log b \)[/tex]
Using this property, we can expand [tex]\( \log \frac{x}{2} \)[/tex]:
[tex]\[ \log \frac{x}{2} = \log x - \log 2 \][/tex]
Now substitute back into the original equation:
[tex]\[ 2 \log x - (\log x - \log 2) = -1 \][/tex]
2. Combine like terms:
Distribute the negative sign and then combine like terms:
[tex]\[ 2 \log x - \log x + \log 2 = -1 \][/tex]
[tex]\[ (2 \log x - \log x) + \log 2 = -1 \][/tex]
[tex]\[ \log x + \log 2 = -1 \][/tex]
3. Simplify further:
Use the logarithm property:
- [tex]\( \log a + \log b = \log (ab) \)[/tex]
Apply this property:
[tex]\[ \log (x \cdot 2) = -1 \][/tex]
[tex]\[ \log (2x) = -1 \][/tex]
4. Solve for [tex]\(x\)[/tex]:
Convert the logarithmic equation into its exponential form. Recall that [tex]\( \log_b (y) = c \)[/tex] is equivalent to [tex]\( y = b^c \)[/tex] (where the base [tex]\( b \)[/tex] is usually 10 for common logarithms or [tex]\( e \)[/tex] for natural logarithms). Here, we are using natural logarithms by default:
[tex]\[ 2x = e^{-1} \][/tex]
Simplify this further:
[tex]\[ 2x = \frac{1}{e} \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{1}{2e} \][/tex]
5. Final Solution:
Let's write the solution in a simplified exponential form:
[tex]\[ x = \frac{e^{-1}}{2} \][/tex]
Hence, the solution to the equation [tex]\( 2 \log x - \log \frac{x}{2} = -1 \)[/tex] is:
[tex]\[ x = \frac{e^{-1}}{2} \][/tex]
