Mara solved an exponential equation by using logarithms. This is her work
1. 7 x 3^x =189
2. 3^x = 27
3.Log _3 3 =9
5. x times 1 = 9
6. x=9
1 Review mara's work in steps 3 and 4 (Hint there is one mistake in those steps
a. Be sure to explain what she did and why it was correct or incorrect for Step 3.
b. Be sure to explain what she did and why it was correct or incorrect for Step 4.
c. Provide Mara with feedback on what the mistake was and how to fix it.
Create your own example of solving an exponential equation, with work shown, that includes at least one mistake that a student could easily make (for example: a common calculation error, or a forgotten or misordered step).
3. Choose another student's problem to review. For each step, explain what they did and why it is correct or incorrect. When you find a mistake, provide feedback to that student as to what mistake(s) you see, what you would have done differently, and why. (If there is no other work to view, use the problem that you wrote. Explain what could be done differently and why that change is necessary to arrive at the correct answer.)
a. In Step 3, Mara wrote "Log _3 3 = 9." This is incorrect. It seems like Mara tried to use the rule that log base b of b is equal to 1. However, in this step, she should have taken the logarithm base 3 of both sides of equation 2, which was 3^x = 27. The correct step would be:
log base 3 of (3^x) = log base 3 of 27
b. In Step 4, Mara wrote "x times 1 = 9." This is also incorrect. In this step, she should have applied the logarithmic property which states that log base b of b^x equals x. Therefore, step 4 should be:
x = log base 3 of 27
c. The mistake that Mara made in step 3 was using the wrong equation to apply the logarithm. The correct equation to apply the logarithm is the one in step 2, which was 3^x = 27. Therefore, step 3 should be:
log base 3 of (3^x) = log base 3 of 27
The mistake Mara made in step 4 was not correctly applying the logarithmic property. The logarithmic property states that log base b of b^x equals x. Therefore, step 4 should be:
x = log base 3 of 27
In this step, Mara needs to evaluate the logarithm of 27 with base 3 using a calculator or by realizing that 3^3 = 27, so log base 3 of 27 is equal to 3.
Therefore, the corrected steps would be:
1. 7 x 3^x = 189
2. 3^x = 27
3. log base 3 of (3^x) = log base 3 of 27
4. x = log base 3 of 27
5. x = 3
Example of solving an exponential equation with a mistake:
Problem: 4 * 2^(2x) = 64
1. 4 * 2^(2x) = 64
2. 2^(2x) = 16
3. log base 2 of (2^(2x)) = log base 2 of 16
4. 2x = log base 2 of 16
5. 2x = 4
Mistake: In step 4, the mistake is not correctly evaluating the logarithm of 16 with base 2. The logarithm of 16 with base 2 is actually 4, not 16.
Corrected steps would be:
1. 4 * 2^(2x) = 64
2. 2^(2x) = 16
3. log base 2 of (2^(2x)) = log base 2 of 16
4. 2x = 4
5. x = 2
Now let's choose another student's problem to review:
Student's problem: 5 * 10^(2x) = 250
1. 5 * 10^(2x) = 250
2. 10^(2x) = 50
3. log base 10 of (10^(2x)) = log base 10 of 50
4. 2x = log base 10 of 50
5. 2x = 1.6990
In this case, the student has correctly applied the logarithmic property and evaluated the logarithm of 50 with base 10. There doesn't seem to be any mistakes in their work. The final answer would be x = 0.8495.