Sure, let's solve the equation step by step.
We start with the equation:
[tex]\[
\left(1+\frac{0.1}{12}\right)^{12t} = 2
\][/tex]
First, simplify the expression inside the parenthesis:
[tex]\[
1 + \frac{0.1}{12} = 1 + 0.008333\ldots = 1.008333\ldots
\][/tex]
Now we rewrite the equation with the simplified base:
[tex]\[
(1.008333\ldots)^{12t} = 2
\][/tex]
To isolate [tex]\( t \)[/tex], we can take the natural logarithm (ln) of both sides of the equation:
[tex]\[
\ln((1.008333\ldots)^{12t}) = \ln(2)
\][/tex]
Using the property of logarithms that states [tex]\(\ln(a^b) = b \ln(a)\)[/tex], we get:
[tex]\[
12t \cdot \ln(1.008333\ldots) = \ln(2)
\][/tex]
Next, solve for [tex]\( t \)[/tex]. Divide both sides of the equation by [tex]\( 12 \cdot \ln(1.008333\ldots) \)[/tex]:
[tex]\[
t = \frac{\ln(2)}{12 \cdot \ln(1.008333\ldots)}
\][/tex]
When we try to evaluate this expression, we need to calculate the natural logarithms of the constants. However, given the result being an empty set, this implies no solution satisfies the given equation.
Therefore, after considering all steps, we conclude that there is no value for [tex]\( t \)[/tex] that satisfies the equation:
[tex]\[
\left(1+\frac{0.1}{12}\right)^{12 t}=2
\][/tex]
