Consider the equation [tex]-3 \cdot 10^{2t} = -28[/tex].

1. Solve the equation for [tex]t[/tex]. Express the solution as a logarithm in base-10.
[tex]\[ t = \boxed{\frac{\log_{10}(\frac{28}{3})}{2}} \][/tex]

2. Approximate the value of [tex]t[/tex]. Round your answer to the nearest thousandth.
[tex]\[ t \approx \boxed{0.649} \][/tex]



Answer :

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]