Stanley Valdez
05/03/2024 · Junior High School
1) Resuelve la siguiente ecuación: \[ \frac{1}{2} \log (2 x-5)-\frac{1}{2} \log (3 x+2)+\log 2=1 \]
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Step-by-step Solution
Solve the equation \( \frac{1}{2} \log (2x-5)-\frac{1}{2} \log (3x+2)+\log 2=1 \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(\frac{1}{2}\log_{10}{\left(2x-5\right)}-\frac{1}{2}\log_{10}{\left(3x+2\right)}+\log_{10}{\left(2\right)}=1\)
- step1: Find the domain:
\(\frac{1}{2}\log_{10}{\left(2x-5\right)}-\frac{1}{2}\log_{10}{\left(3x+2\right)}+\log_{10}{\left(2\right)}=1,x>\frac{5}{2}\)
- step2: Subtract the terms:
\(\frac{1}{2}\log_{10}{\left(\frac{2x-5}{3x+2}\right)}+\log_{10}{\left(2\right)}=1\)
- step3: Move the expression to the left side:
\(\frac{1}{2}\log_{10}{\left(\frac{2x-5}{3x+2}\right)}+\log_{10}{\left(2\right)}-1=0\)
- step4: Solve using substitution:
\(\frac{1}{2}t+\log_{10}{\left(2\right)}-1=0\)
- step5: Move the constant to the right side:
\(\frac{1}{2}t=0-\left(\log_{10}{\left(2\right)}-1\right)\)
- step6: Subtract the terms:
\(\frac{1}{2}t=-\log_{10}{\left(2\right)}+1\)
- step7: Multiply by the reciprocal:
\(\frac{1}{2}t\times 2=\left(-\log_{10}{\left(2\right)}+1\right)\times 2\)
- step8: Multiply:
\(t=-2\log_{10}{\left(2\right)}+2\)
- step9: Substitute back:
\(\log_{10}{\left(\frac{2x-5}{3x+2}\right)}=-2\log_{10}{\left(2\right)}+2\)
- step10: Evaluate the logarithm:
\(\log_{10}{\left(\frac{2x-5}{3x+2}\right)}=\log_{10}{\left(25\right)}\)
- step11: Evaluate the logarithm:
\(\frac{2x-5}{3x+2}=25\)
- step12: Cross multiply:
\(2x-5=\left(3x+2\right)\times 25\)
- step13: Simplify the equation:
\(2x-5=25\left(3x+2\right)\)
- step14: Expand the expression:
\(2x-5=75x+50\)
- step15: Move the variable to the left side:
\(2x-5-75x=50\)
- step16: Subtract the terms:
\(-73x-5=50\)
- step17: Move the constant to the right side:
\(-73x=50+5\)
- step18: Add the numbers:
\(-73x=55\)
- step19: Change the signs:
\(73x=-55\)
- step20: Divide both sides:
\(\frac{73x}{73}=\frac{-55}{73}\)
- step21: Divide the numbers:
\(x=-\frac{55}{73}\)
- step22: Check if the solution is in the defined range:
\(x=-\frac{55}{73},x>\frac{5}{2}\)
- step23: Find the intersection:
\(x \in \varnothing \)
La solución de la ecuación dada es \( x \in \varnothing \), lo que significa que no hay solución real para esta ecuación.
Quick Answer
La ecuación no tiene solución real.
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