To solve the exponential equation [tex]\( 10e^x - 2 = 8 \)[/tex], we can follow a series of logical steps:
1. Isolate the exponential term:
[tex]\[
10e^x - 2 = 8
\][/tex]
Add 2 to both sides to isolate the term involving [tex]\( e^x \)[/tex]:
[tex]\[
10e^x = 10
\][/tex]
2. Solve for [tex]\( e^x \)[/tex]:
Divide both sides by 10 to further isolate [tex]\( e^x \)[/tex]:
[tex]\[
e^x = 1
\][/tex]
3. Solve for [tex]\( x \)[/tex] using the natural logarithm:
Since [tex]\( e^x = 1 \)[/tex], we can take the natural logarithm of both sides:
[tex]\[
\ln(e^x) = \ln(1)
\][/tex]
By properties of logarithms, we know that [tex]\( \ln(e^x) = x \cdot \ln(e) \)[/tex] and [tex]\( \ln(e) = 1 \)[/tex], hence:
[tex]\[
x = \ln(1)
\][/tex]
We also know that [tex]\( \ln(1) = 0 \)[/tex], therefore:
[tex]\[
x = 0
\][/tex]
Thus, in terms of logarithms, the solution for [tex]\( x \)[/tex] is [tex]\( \ln(1) \)[/tex], and the exact value is [tex]\( x = 0 \)[/tex].
Finally, to confirm the solution:
- In terms of logarithms: [tex]\( x = \ln(1) \)[/tex]
- Exact numerical value: [tex]\( x = 0.0 \)[/tex] (correct to four decimal places)
