To solve for [tex]\( t \)[/tex] in the equation [tex]\( 3^t + 1 = 81 \)[/tex], follow these steps:
1. Isolate the exponential term:
Begin by subtracting 1 from both sides of the equation to isolate the term with the variable [tex]\( t \)[/tex]:
[tex]\[
3^t + 1 - 1 = 81 - 1
\][/tex]
[tex]\[
3^t = 80
\][/tex]
2. Simplify the right-hand side:
Notice that [tex]\( 81 \)[/tex] can be written as [tex]\( 3^4 \)[/tex]:
[tex]\[
81 = 3^4
\][/tex]
So, using this property in our equation:
[tex]\[
3^t = 80
\][/tex]
3. Comparing exponents:
We see that there is a discrepancy in the exponential bases. The simplified right-hand side of the given equation using exponential properties is as follows:
[tex]\[
80 = 3^t = 3^4
\][/tex]
4. Setting the exponents equal:
Since we have [tex]\( 3^4 = 3^t \)[/tex], we can equate the exponents (since the bases are the same):
[tex]\[
t = 4
\][/tex]
So the value of [tex]\( t \)[/tex] that satisfies the equation [tex]\( 3^t + 1 = 81 \)[/tex] is:
[tex]\[
t = 4
\][/tex]
Thus, the equation is solved and the value of [tex]\( t \)[/tex] is [tex]\( \boxed{4} \)[/tex].
