Let's solve each of the equations step-by-step:
### i. [tex]\( 2^x = 10 \)[/tex]
1. Take the natural logarithm of both sides:
[tex]\[
\ln(2^x) = \ln(10)
\][/tex]
2. Utilize the property of logarithms: [tex]\(\ln(a^b) = b \ln(a)\)[/tex]:
[tex]\[
x \cdot \ln(2) = \ln(10)
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{\ln(10)}{\ln(2)}
\][/tex]
4. Approximate the values of the logarithms:
[tex]\[
\ln(10) \approx 2.302585
\][/tex]
[tex]\[
\ln(2) \approx 0.693147
\][/tex]
5. Perform the division:
[tex]\[
x \approx \frac{2.302585}{0.693147} \approx 3.321928
\][/tex]
So, the solution to the first equation is:
[tex]\[
x \approx 3.321928
\][/tex]
### ii. [tex]\( 5^{4x-1} = 7^{x+1} \)[/tex]
1. Take the natural logarithm of both sides:
[tex]\[
\ln(5^{4x-1}) = \ln(7^{x+1})
\][/tex]
2. Utilize the property of logarithms: [tex]\(\ln(a^b) = b \ln(a)\)[/tex]:
[tex]\[
(4x - 1) \cdot \ln(5) = (x + 1) \cdot \ln(7)
\][/tex]
3. Distribute the logarithms:
[tex]\[
4x \cdot \ln(5) - \ln(5) = x \cdot \ln(7) + \ln(7)
\][/tex]
4. Rearrange to isolate terms involving [tex]\(x\)[/tex]:
[tex]\[
4x \cdot \ln(5) - x \cdot \ln(7) = \ln(7) + \ln(5)
\][/tex]
5. Factor out [tex]\(x\)[/tex] on the left-hand side:
[tex]\[
x \cdot (4 \ln(5) - \ln(7)) = \ln(7) + \ln(5)
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{\ln(7) + \ln(5)}{4 \ln(5) - \ln(7)}
\][/tex]
7. Approximate the values of the logarithms:
[tex]\[
\ln(5) \approx 1.609438
\][/tex]
[tex]\[
\ln(7) \approx 1.945910
\][/tex]
8. Substitute the approximate values:
[tex]\[
x \approx \frac{1.945910 + 1.609438}{4 \cdot 1.609438 - 1.945910}
\][/tex]
9. Perform the arithmetic:
[tex]\[
x \approx \frac{3.555348}{4 \cdot 1.609438 - 1.945910} \approx \frac{3.555348}{6.437752 - 1.945910} \approx \frac{3.555348}{4.491842} \approx 0.791512
\][/tex]
So, the solution to the second equation is:
[tex]\[
x \approx 0.791512
\][/tex]
### Summary
The solutions to the given equations are approximately:
i. [tex]\( x \approx 3.321928 \)[/tex]
ii. [tex]\( x \approx 0.791512 \)[/tex]
