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\( x \approx 129 \) and \( y \approx 0.349 \).
Solución
Solve the equation by following steps:
- step0: Solve for \(y\):
\(2^{y+1}=\frac{10}{2^{y}}\)
- step1: Cross multiply:
\(2^{y+1}\times 2^{y}=10\)
- step2: Simplify the equation:
\(2^{2y+1}=10\)
- step3: Take the logarithm of both sides:
\(\log_{2}{\left(2^{2y+1}\right)}=\log_{2}{\left(10\right)}\)
- step4: Evaluate the logarithm:
\(2y+1=\log_{2}{\left(10\right)}\)
- step5: Move the constant to the right side:
\(2y=\log_{2}{\left(10\right)}-1\)
- step6: Divide both sides:
\(\frac{2y}{2}=\frac{\log_{2}{\left(10\right)}-1}{2}\)
- step7: Divide the numbers:
\(y=\frac{\log_{2}{\left(10\right)}-1}{2}\)
- step8: Simplify:
\(y=\frac{\log_{2}{\left(5\right)}}{2}\)
Solve the equation \( 5^{2x-1}=4^{300} \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(5^{2x-1}=4^{300}\)
- step1: Take the logarithm of both sides:
\(\log_{5}{\left(5^{2x-1}\right)}=\log_{5}{\left(4^{300}\right)}\)
- step2: Evaluate the logarithm:
\(2x-1=\log_{5}{\left(4^{300}\right)}\)
- step3: Simplify:
\(2x-1=600\log_{5}{\left(2\right)}\)
- step4: Move the constant to the right side:
\(2x=600\log_{5}{\left(2\right)}+1\)
- step5: Divide both sides:
\(\frac{2x}{2}=\frac{600\log_{5}{\left(2\right)}+1}{2}\)
- step6: Divide the numbers:
\(x=\frac{600\log_{5}{\left(2\right)}+1}{2}\)
Calculate or simplify the expression \( \log(5)/2 \).
Calculate the value by following steps:
- step0: Calculate:
\(\frac{\log_{10}{\left(5\right)}}{2}\)
Calculate or simplify the expression \( (600*\log(2)/\log(5)+1)/2 \).
Calculate the value by following steps:
- step0: Calculate:
\(\frac{\left(\frac{600\log_{10}{\left(2\right)}}{\log_{10}{\left(5\right)}}+1\right)}{2}\)
- step1: Remove the parentheses:
\(\frac{\frac{600\log_{10}{\left(2\right)}}{\log_{10}{\left(5\right)}}+1}{2}\)
- step2: Use the logarithm base change rule:
\(\frac{600\log_{5}{\left(2\right)}+1}{2}\)
Let's solve the given exponential equations step by step.
### i. Solve \( 5^{2x-1} = 4^{300} \)
1. **Rewrite the equation**:
\[
5^{2x-1} = (2^2)^{300} = 2^{600}
\]
2. **Take logarithm on both sides**:
\[
(2x - 1) \log(5) = 600 \log(2)
\]
3. **Isolate \( x \)**:
\[
2x - 1 = \frac{600 \log(2)}{\log(5)}
\]
\[
2x = \frac{600 \log(2)}{\log(5)} + 1
\]
\[
x = \frac{600 \log(2)}{2 \log(5)} + \frac{1}{2}
\]
4. **Calculate the value**:
\[
x \approx 129.703 \quad (\text{to 3 significant figures: } 129)
\]
### ii. Solve \( 2^{y+1} = \frac{10}{2^{y}} \)
1. **Rewrite the equation**:
\[
2^{y+1} \cdot 2^{y} = 10
\]
\[
2^{2y+1} = 10
\]
2. **Take logarithm on both sides**:
\[
(2y + 1) \log(2) = \log(10)
\]
3. **Isolate \( y \)**:
\[
2y + 1 = \frac{\log(10)}{\log(2)}
\]
\[
2y = \frac{\log(10)}{\log(2)} - 1
\]
\[
y = \frac{\log(10)}{2 \log(2)} - \frac{1}{2}
\]
4. **Calculate the value**:
\[
y \approx 0.349 \quad (\text{to 3 significant figures: } 0.349)
\]
### Final Answers:
- \( x \approx 129 \)
- \( y \approx 0.349 \)
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