To find the value of [tex]\( t \)[/tex] in the given equation [tex]\( \frac{1}{4}(10)^{2t} = 5 \)[/tex], follow these steps:
1. Isolate the Exponential Term:
Start by isolating the exponential term [tex]\((10)^{2t}\)[/tex]. To do this, multiply both sides of the equation by 4:
[tex]\[
4 \cdot \frac{1}{4}(10)^{2t} = 5 \cdot 4
\][/tex]
Simplifying the left side, we get:
[tex]\[
(10)^{2t} = 20
\][/tex]
2. Apply the Logarithm:
Apply the logarithm on both sides of the equation. Common logarithms (base 10) are typically used for problems involving base 10 exponentials:
[tex]\[
\log_{10}((10)^{2t}) = \log_{10}(20)
\][/tex]
Using the property of logarithms [tex]\(\log_{10}(a^b) = b \cdot \log_{10}(a)\)[/tex], we can simplify the left side:
[tex]\[
2t \cdot \log_{10}(10) = \log_{10}(20)
\][/tex]
3. Simplify the Equation:
Since [tex]\(\log_{10}(10) = 1\)[/tex], the equation simplifies further:
[tex]\[
2t \cdot 1 = \log_{10}(20)
\][/tex]
[tex]\[
2t = \log_{10}(20)
\][/tex]
4. Solve for [tex]\( t \)[/tex]:
To solve for [tex]\( t \)[/tex], divide both sides by 2:
[tex]\[
t = \frac{\log_{10}(20)}{2}
\][/tex]
5. Numerical Value:
Use a calculator to find the logarithm value:
[tex]\[
\log_{10}(20) \approx 1.30103
\][/tex]
Divide this value by 2:
[tex]\[
t = \frac{1.30103}{2} \approx 0.6505149978319906
\][/tex]
6. Rounding:
Round the answer to the nearest hundredth:
[tex]\[
t \approx 0.65
\][/tex]
Therefore, the value of [tex]\( t \)[/tex] rounded to the nearest hundredth is:
[tex]\[
\boxed{0.65}
\][/tex]
