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Answer :
To solve the equation [tex]\(\log_e 2 \cdot \log_b 625 = \log_{10} 16 \cdot \log_e 10\)[/tex], we need to find the value of [tex]\(b\)[/tex].
Step 1: Understand the equation
The given equation is:
[tex]\[ \log_e 2 \cdot \log_b 625 = \log_{10} 16 \cdot \log_e 10 \][/tex]
Step 2: Apply the change of base formula
We can use the change of base formula which states:
[tex]\[
\log_b x = \frac{\log_e x}{\log_e b}
\][/tex]
Step 3: Rewrite the equation using natural logarithms
Using the change of base formula, we have:
[tex]\[ \log_b 625 = \frac{\log_e 625}{\log_e b} \][/tex]
[tex]\[ \log_{10} 16 = \frac{\log_e 16}{\log_e 10} \][/tex]
Replace these in the original equation:
[tex]\[
\log_e 2 \cdot \frac{\log_e 625}{\log_e b} = \frac{\log_e 16}{\log_e 10} \cdot \log_e 10
\][/tex]
Which simplifies to:
[tex]\[
\log_e 2 \cdot \frac{\log_e 625}{\log_e b} = \log_e 16
\][/tex]
Step 4: Solve for [tex]\(\log_e b\)[/tex]
This simplifies the equation to:
[tex]\[
\log_e b = \frac{\log_e 625 \cdot \log_e 2}{\log_e 16}
\][/tex]
Step 5: Calculate individual logs
Now, calculate each component:
- [tex]\(\log_e 2\)[/tex]
- [tex]\(\log_e 625\)[/tex]
- [tex]\(\log_e 16\)[/tex]
From the Python solution:
- [tex]\(\log_e 2 \approx 0.693147\)[/tex]
- [tex]\(\log_e 625 \approx 6.437752\)[/tex]
- [tex]\(\log_e 16 \approx 2.772589\)[/tex]
Step 6: Substitute known values and solve
Substitute these values back into our expression for [tex]\(\log_e b\)[/tex]:
[tex]\[
\log_e b \approx \frac{6.437752 \times 0.693147}{2.772589}
\][/tex]
When you do this calculation, you find:
[tex]\(\log_e b \approx 1.609438\)[/tex]
Step 7: Calculate [tex]\(b\)[/tex]
Now, solve for [tex]\(b\)[/tex] by using the exponential function:
[tex]\[
b = e^{\log_e b} \approx e^{1.609438} = 5
\][/tex]
Thus, the value of [tex]\(b\)[/tex] is [tex]\(5\)[/tex].
Step 1: Understand the equation
The given equation is:
[tex]\[ \log_e 2 \cdot \log_b 625 = \log_{10} 16 \cdot \log_e 10 \][/tex]
Step 2: Apply the change of base formula
We can use the change of base formula which states:
[tex]\[
\log_b x = \frac{\log_e x}{\log_e b}
\][/tex]
Step 3: Rewrite the equation using natural logarithms
Using the change of base formula, we have:
[tex]\[ \log_b 625 = \frac{\log_e 625}{\log_e b} \][/tex]
[tex]\[ \log_{10} 16 = \frac{\log_e 16}{\log_e 10} \][/tex]
Replace these in the original equation:
[tex]\[
\log_e 2 \cdot \frac{\log_e 625}{\log_e b} = \frac{\log_e 16}{\log_e 10} \cdot \log_e 10
\][/tex]
Which simplifies to:
[tex]\[
\log_e 2 \cdot \frac{\log_e 625}{\log_e b} = \log_e 16
\][/tex]
Step 4: Solve for [tex]\(\log_e b\)[/tex]
This simplifies the equation to:
[tex]\[
\log_e b = \frac{\log_e 625 \cdot \log_e 2}{\log_e 16}
\][/tex]
Step 5: Calculate individual logs
Now, calculate each component:
- [tex]\(\log_e 2\)[/tex]
- [tex]\(\log_e 625\)[/tex]
- [tex]\(\log_e 16\)[/tex]
From the Python solution:
- [tex]\(\log_e 2 \approx 0.693147\)[/tex]
- [tex]\(\log_e 625 \approx 6.437752\)[/tex]
- [tex]\(\log_e 16 \approx 2.772589\)[/tex]
Step 6: Substitute known values and solve
Substitute these values back into our expression for [tex]\(\log_e b\)[/tex]:
[tex]\[
\log_e b \approx \frac{6.437752 \times 0.693147}{2.772589}
\][/tex]
When you do this calculation, you find:
[tex]\(\log_e b \approx 1.609438\)[/tex]
Step 7: Calculate [tex]\(b\)[/tex]
Now, solve for [tex]\(b\)[/tex] by using the exponential function:
[tex]\[
b = e^{\log_e b} \approx e^{1.609438} = 5
\][/tex]
Thus, the value of [tex]\(b\)[/tex] is [tex]\(5\)[/tex].
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