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Answer :
Certainly! Let's solve each equation step-by-step:
Equation 45: [tex]\(\log(7x + 6) = 3\)[/tex]
1. Understand the logarithmic equation: The equation [tex]\(\log(7x + 6) = 3\)[/tex] uses the common logarithm (base 10).
2. Convert to exponential form: From the property of logarithms, if [tex]\(\log_{10}(a) = c\)[/tex], then [tex]\(a = 10^c\)[/tex]. Here, that means:
[tex]\[
7x + 6 = 10^3
\][/tex]
3. Calculate [tex]\(10^3\)[/tex]:
[tex]\[
10^3 = 1000
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
[tex]\[
7x + 6 = 1000
\][/tex]
Subtract 6 from both sides:
[tex]\[
7x = 994
\][/tex]
5. Divide by 7 to isolate [tex]\(x\)[/tex]:
[tex]\[
x = \frac{994}{7}
\][/tex]
[tex]\[
x = 142.0
\][/tex]
So, the solution to the first equation is [tex]\(x = 142.0\)[/tex].
---
Equation 46: [tex]\(2.75 \cdot e^t = 38.6\)[/tex]
1. Isolate the exponential expression: Divide both sides by 2.75:
[tex]\[
e^t = \frac{38.6}{2.75}
\][/tex]
2. Calculate the division:
[tex]\[
e^t \approx 14.0364
\][/tex]
3. Solve for [tex]\(t\)[/tex] using the natural logarithm: If [tex]\(e^t = a\)[/tex], then [tex]\(t = \ln(a)\)[/tex].
[tex]\[
t = \ln(14.0364)
\][/tex]
4. Calculate the natural logarithm:
[tex]\[
t \approx 2.6417
\][/tex]
So, the solution to the second equation is [tex]\(t \approx 2.6417\)[/tex].
These are the solutions for the equations:
- For equation 45: [tex]\(x = 142.0\)[/tex]
- For equation 46: [tex]\(t \approx 2.6417\)[/tex]
Equation 45: [tex]\(\log(7x + 6) = 3\)[/tex]
1. Understand the logarithmic equation: The equation [tex]\(\log(7x + 6) = 3\)[/tex] uses the common logarithm (base 10).
2. Convert to exponential form: From the property of logarithms, if [tex]\(\log_{10}(a) = c\)[/tex], then [tex]\(a = 10^c\)[/tex]. Here, that means:
[tex]\[
7x + 6 = 10^3
\][/tex]
3. Calculate [tex]\(10^3\)[/tex]:
[tex]\[
10^3 = 1000
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
[tex]\[
7x + 6 = 1000
\][/tex]
Subtract 6 from both sides:
[tex]\[
7x = 994
\][/tex]
5. Divide by 7 to isolate [tex]\(x\)[/tex]:
[tex]\[
x = \frac{994}{7}
\][/tex]
[tex]\[
x = 142.0
\][/tex]
So, the solution to the first equation is [tex]\(x = 142.0\)[/tex].
---
Equation 46: [tex]\(2.75 \cdot e^t = 38.6\)[/tex]
1. Isolate the exponential expression: Divide both sides by 2.75:
[tex]\[
e^t = \frac{38.6}{2.75}
\][/tex]
2. Calculate the division:
[tex]\[
e^t \approx 14.0364
\][/tex]
3. Solve for [tex]\(t\)[/tex] using the natural logarithm: If [tex]\(e^t = a\)[/tex], then [tex]\(t = \ln(a)\)[/tex].
[tex]\[
t = \ln(14.0364)
\][/tex]
4. Calculate the natural logarithm:
[tex]\[
t \approx 2.6417
\][/tex]
So, the solution to the second equation is [tex]\(t \approx 2.6417\)[/tex].
These are the solutions for the equations:
- For equation 45: [tex]\(x = 142.0\)[/tex]
- For equation 46: [tex]\(t \approx 2.6417\)[/tex]
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