Question
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.
Answers
GPT 3.5
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.
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.