To solve the equation [tex]\(-3 \cdot 10^{2 t} = -28\)[/tex] for [tex]\(t\)[/tex], follow these steps:
1. Isolate the exponential term:
[tex]\[
-3 \cdot 10^{2 t} = -28
\][/tex]
Divide both sides of the equation by [tex]\(-3\)[/tex]:
[tex]\[
10^{2 t} = \frac{28}{3}
\][/tex]
2. Take the logarithm on both sides:
To solve for [tex]\(t\)[/tex], take the base-10 logarithm of both sides:
[tex]\[
\log_{10}(10^{2t}) = \log_{10}\left(\frac{28}{3}\right)
\][/tex]
3. Simplify using logarithm properties:
We know that [tex]\(\log_{10}(10^{2t}) = 2t\)[/tex], so the equation becomes:
[tex]\[
2t = \log_{10}\left(\frac{28}{3}\right)
\][/tex]
4. Solve for [tex]\(t\)[/tex]:
Divide both sides of the equation by 2:
[tex]\[
t = \frac{1}{2} \log_{10}\left(\frac{28}{3}\right)
\][/tex]
5. Express the exact solution:
The exact value of [tex]\(t\)[/tex] can be written as:
[tex]\[
t = \frac{1}{2} \log_{10}\left(\frac{28}{3}\right)
\][/tex]
6. Approximate the value of [tex]\(t\)[/tex]:
Using a calculator, find the logarithm in base-10 of [tex]\(\frac{28}{3}\)[/tex]:
[tex]\[
\log_{10}\left(\frac{28}{3}\right) \approx 0.9700367766225568
\][/tex]
Then multiply by [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[
t \approx \frac{1}{2} \times 0.9700367766225568 \approx 0.4850183883112784
\][/tex]
7. Round to the nearest thousandth:
[tex]\[
t \approx 0.485
\][/tex]
So, the exact solution is:
[tex]\[
t = \frac{1}{2} \log_{10}\left(\frac{28}{3}\right)
\][/tex]
And the approximate value, rounded to the nearest thousandth, is:
[tex]\[
t \approx 0.485
\][/tex]
