Find the solution of the exponential equation:

[tex]\[ 10e^x - 2 = 8 \][/tex]

in terms of logarithms, or correct to four decimal places.

[tex]\[ x = \square \][/tex]



Answer :

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)