Answered

Discover new knowledge and insights with IDNLearn.com's extensive Q&A database. Join our community to receive prompt and reliable responses to your questions from knowledgeable professionals.

Solve the exponential equation. Write the exact solution and the approximation to four decimal places. Type "DNE" if there is no solution.

[tex]\[2 \cdot 8^{x-3} = 7\][/tex]

1. Exact solution: [tex]\(\square\)[/tex]

2. Approximation: [tex]\(\square\)[/tex]

Sagot :

GinnyAnsweridn
To solve the exponential equation [tex]\( 2 \cdot 8^{x-3} = 7 \)[/tex], we need to follow these steps:

### Step-by-Step Solution:
1. Isolate the exponential term:
[tex]\[ 8^{x-3} = \frac{7}{2} \][/tex]

2. Take the logarithm of both sides:
[tex]\[ \log\left(8^{x-3}\right) = \log\left(\frac{7}{2}\right) \][/tex]

3. Use the power rule of logarithms: The power rule states that [tex]\(\log(a^b) = b \cdot \log(a)\)[/tex].
[tex]\[ (x-3) \cdot \log(8) = \log\left(\frac{7}{2}\right) \][/tex]

4. Solve for [tex]\(x\)[/tex]:
[tex]\[ x - 3 = \frac{\log\left(\frac{7}{2}\right)}{\log(8)} \][/tex]
[tex]\[ x = \frac{\log\left(\frac{7}{2}\right)}{\log(8)} + 3 \][/tex]

5. Determine the exact solution:
[tex]\[ x = \frac{\log\left(\frac{7}{2}\right)}{\log(8)} + 3 \][/tex]

6. Approximate [tex]\(\frac{\log\left(\frac{7}{2}\right)}{\log(8)}\)[/tex] to 4 decimal places:
[tex]\[ \frac{\log\left(3.5\right)}{\log(8)} \approx 0.6024 \][/tex]
Adding 3:
[tex]\[ x \approx 0.6024 + 3 = 3.6024 \][/tex]

### Final Answers:
1. Exact solution:
[tex]\[ x = \frac{\log\left(\frac{7}{2}\right)}{\log(8)} + 3 \][/tex]

2. Approximation to 4 decimal places:
[tex]\[ x \approx 3.6024 \][/tex]
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. IDNLearn.com is your go-to source for dependable answers. Thank you for visiting, and we hope to assist you again.