The exponential equation 3^x=12 does not have an integer solution. Use an over/under table to find an estimate of the solution to two decimal places.
To find an estimate of the solution to two decimal places, we can use an over/under table.
First, let's find the values of 3^x for various integer values of x:
x | 3^x
------------
-5 | 1/243
-4 | 1/81
-3 | 1/27
-2 | 1/9
-1 | 1/3
0 | 1
1 | 3
2 | 9
3 | 27
4 | 81
5 | 243
From the table, we can see that when x is negative, 3^x gets smaller and smaller. On the other hand, when x is positive, 3^x gets larger and larger.
Since we want to find a solution to the equation 3^x = 12, we can first compare 3^x to 12. From the table, we can see that 3^2 = 9, which is less than 12, and 3^3 = 27, which is greater than 12. This suggests that the solution lies between x = 2 and x = 3.
To get a more accurate estimate, we can use a linear interpolation. We calculate the difference between 3^x and 12 for x = 2 and x = 3:
When x = 2:
Difference = 12 - 9 = 3
When x = 3:
Difference = 27 - 12 = 15
The difference between x = 2 and x = 3 is 1.
To estimate the solution, we can calculate the fraction of the difference between x = 2 and x = 3 that corresponds to the difference between 3^x and 12:
Fraction of difference = Difference / (Difference between x = 2 and x = 3)
Fraction of difference = 3 / 1 = 3
To find the estimated solution, we add the fraction of difference to x = 2:
Estimated solution = x = 2 + Fraction of difference
Estimated solution = 2 + 3 = 5
Therefore, the estimated solution to two decimal places is x = 5.