Fravel, Dan, Math / CP-Algebra 2 - Free Printable
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Step-by-step solution for: Fravel, Dan, Math / CP-Algebra 2
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Show Answer Key & Explanations
Step-by-step solution for: Fravel, Dan, Math / CP-Algebra 2
Let’s go through each problem one by one. These are all about logarithms and exponents — basically, they’re asking you to find powers or reverse them.
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1. log₂8 = 3
Because 2³ = 8 → correct.
2. log₃81 = 4
Because 3⁴ = 81 → correct.
3. log₄64 = 3
Because 4³ = 64 → correct.
4. log₅625 = 4
Because 5⁴ = 625 → correct.
5. log 10,000 = 4
This is base 10 (common log). 10⁴ = 10,000 → correct.
6. log₂(1/8) = -3
Because 2⁻³ = 1/(2³) = 1/8 → correct.
7. log₂₇9 = 2/3
We need: 27^(?) = 9
27 = 3³, 9 = 3² → so (3³)^(2/3) = 3² = 9 → correct.
8. log₁₆8 = 3/4
16 = 2⁴, 8 = 2³ → (2⁴)^(3/4) = 2³ = 8 → correct.
9. 2⁴ = 16 → correct.
10. 3⁻³ = 1/27 → because 3³=27, negative exponent means reciprocal → correct.
11. 10² = 100 → correct.
12. 5³ = 125 → correct.
13. 8² = 64 → correct.
14. 9¹ = 9 → correct.
15. 12⁰ = 1 → any number to power 0 is 1 → correct.
16. 23⁰ = 1 → same rule → correct.
17. 5⁻¹ = 1/5 → correct.
18. 7¹ = 7 → correct.
19. 2⁶ = 64 → 2×2×2×2×2×2 = 64 → correct.
20. 6² = 36 → correct.
Now problems 21–38 seem to be answers to other questions not shown — but since the user just gave this list, I’ll assume we’re verifying if these are correct as written.
Looking at 21–38:
These look like standalone numbers — possibly answers to missing questions? But since no context is given for what question they answer, we can’t verify them unless they’re meant to be evaluated as-is.
Wait — looking again, maybe 21–38 are *answers* to previous problems? But problems 1–20 already have answers next to them.
Actually, re-reading the image text — it says “Practice 7-3: Example Exercises” and lists 1–38 with values after equals signs. So perhaps 21–38 are also equations where the left side is missing? That doesn’t make sense.
Alternatively — maybe 21–38 are just numerical answers to unlisted problems? Since the instruction is to “solve the problem accurately”, and the only visible content is this list, I think the intent is to check which of these statements are true.
But 21–38 don’t have operators — they’re just numbers. For example:
21. 2 → that’s not an equation.
Unless... perhaps in the original worksheet, 21–38 were meant to be filled-in answers, and here they’re provided as solutions? But without the questions, we can’t validate them.
Given the ambiguity, and since problems 1–20 are complete equations and all are mathematically correct, while 21–38 are just numbers (not equations), I will assume the task is to confirm the correctness of 1–20, and note that 21–38 lack context.
But wait — let me double-check problem 7 and 8 again, since fractions in logs can be tricky.
Problem 7: log₂₇9 = 2/3
Let x = log₂₇9 → then 27^x = 9
27 = 3³, 9 = 3² → (3³)^x = 3² → 3^(3x) = 3² → 3x = 2 → x = 2/3 → ✔ Correct.
Problem 8: log₁₆8 = 3/4
16^x = 8 → (2⁴)^x = 2³ → 2^(4x) = 2³ → 4x = 3 → x = 3/4 → ✔ Correct.
All others are straightforward exponent rules.
So problems 1–20 are all correctly solved.
Problems 21–38: Since they are just numbers without operations or context, they cannot be verified as “correct” or “incorrect” without knowing what question they’re answering. However, if we assume they are meant to be final answers to some implied calculations, and since no calculation is shown, we must treat them as given.
But the user said “Solve the problem accurately.” — implying there might be errors to fix? Or just to confirm?
Looking back — actually, in many textbooks, such lists are “answer keys” — meaning these are the correct answers to exercises. So perhaps the student is supposed to match or use them.
Since no specific question was asked beyond “solve the problem”, and the image shows a list of equations and values, I believe the intended task is to recognize that all equations from 1 to 20 are correct, and 21–38 are likely answer values for other problems not shown.
Therefore, if forced to give a “final answer” based on what’s presented, I’d say:
All given logarithmic and exponential equations (problems 1–20) are mathematically correct.
But since the format requires a single final answer, and considering that sometimes such practice sheets ask students to identify incorrect ones — but none are incorrect — perhaps the answer is simply that everything checks out.
Alternatively, maybe problem 27 is written as “½” which is 0.5, and others are decimals — but again, no issue.
Wait — problem 37: 4.0 × 10⁻² = 0.04 — that’s fine.
Problem 38: 1.0 × 10⁻³ = 0.001 — fine.
I think the safest conclusion is: All stated equalities in problems 1–20 are correct. Problems 21–38 appear to be numerical answers without corresponding questions, so their validity cannot be assessed here.
But since the user expects a final answer, and given that this is labeled “Example Exercises” with solutions provided, perhaps the point is to accept them as correct examples.
Final decision: Since no errors exist in 1–20, and 21–38 are not equations, I will state that the provided solutions for 1–20 are accurate.
Final Answer:
All equations from 1 to 20 are correct.
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1. log₂8 = 3
Because 2³ = 8 → correct.
2. log₃81 = 4
Because 3⁴ = 81 → correct.
3. log₄64 = 3
Because 4³ = 64 → correct.
4. log₅625 = 4
Because 5⁴ = 625 → correct.
5. log 10,000 = 4
This is base 10 (common log). 10⁴ = 10,000 → correct.
6. log₂(1/8) = -3
Because 2⁻³ = 1/(2³) = 1/8 → correct.
7. log₂₇9 = 2/3
We need: 27^(?) = 9
27 = 3³, 9 = 3² → so (3³)^(2/3) = 3² = 9 → correct.
8. log₁₆8 = 3/4
16 = 2⁴, 8 = 2³ → (2⁴)^(3/4) = 2³ = 8 → correct.
9. 2⁴ = 16 → correct.
10. 3⁻³ = 1/27 → because 3³=27, negative exponent means reciprocal → correct.
11. 10² = 100 → correct.
12. 5³ = 125 → correct.
13. 8² = 64 → correct.
14. 9¹ = 9 → correct.
15. 12⁰ = 1 → any number to power 0 is 1 → correct.
16. 23⁰ = 1 → same rule → correct.
17. 5⁻¹ = 1/5 → correct.
18. 7¹ = 7 → correct.
19. 2⁶ = 64 → 2×2×2×2×2×2 = 64 → correct.
20. 6² = 36 → correct.
Now problems 21–38 seem to be answers to other questions not shown — but since the user just gave this list, I’ll assume we’re verifying if these are correct as written.
Looking at 21–38:
These look like standalone numbers — possibly answers to missing questions? But since no context is given for what question they answer, we can’t verify them unless they’re meant to be evaluated as-is.
Wait — looking again, maybe 21–38 are *answers* to previous problems? But problems 1–20 already have answers next to them.
Actually, re-reading the image text — it says “Practice 7-3: Example Exercises” and lists 1–38 with values after equals signs. So perhaps 21–38 are also equations where the left side is missing? That doesn’t make sense.
Alternatively — maybe 21–38 are just numerical answers to unlisted problems? Since the instruction is to “solve the problem accurately”, and the only visible content is this list, I think the intent is to check which of these statements are true.
But 21–38 don’t have operators — they’re just numbers. For example:
21. 2 → that’s not an equation.
Unless... perhaps in the original worksheet, 21–38 were meant to be filled-in answers, and here they’re provided as solutions? But without the questions, we can’t validate them.
Given the ambiguity, and since problems 1–20 are complete equations and all are mathematically correct, while 21–38 are just numbers (not equations), I will assume the task is to confirm the correctness of 1–20, and note that 21–38 lack context.
But wait — let me double-check problem 7 and 8 again, since fractions in logs can be tricky.
Problem 7: log₂₇9 = 2/3
Let x = log₂₇9 → then 27^x = 9
27 = 3³, 9 = 3² → (3³)^x = 3² → 3^(3x) = 3² → 3x = 2 → x = 2/3 → ✔ Correct.
Problem 8: log₁₆8 = 3/4
16^x = 8 → (2⁴)^x = 2³ → 2^(4x) = 2³ → 4x = 3 → x = 3/4 → ✔ Correct.
All others are straightforward exponent rules.
So problems 1–20 are all correctly solved.
Problems 21–38: Since they are just numbers without operations or context, they cannot be verified as “correct” or “incorrect” without knowing what question they’re answering. However, if we assume they are meant to be final answers to some implied calculations, and since no calculation is shown, we must treat them as given.
But the user said “Solve the problem accurately.” — implying there might be errors to fix? Or just to confirm?
Looking back — actually, in many textbooks, such lists are “answer keys” — meaning these are the correct answers to exercises. So perhaps the student is supposed to match or use them.
Since no specific question was asked beyond “solve the problem”, and the image shows a list of equations and values, I believe the intended task is to recognize that all equations from 1 to 20 are correct, and 21–38 are likely answer values for other problems not shown.
Therefore, if forced to give a “final answer” based on what’s presented, I’d say:
All given logarithmic and exponential equations (problems 1–20) are mathematically correct.
But since the format requires a single final answer, and considering that sometimes such practice sheets ask students to identify incorrect ones — but none are incorrect — perhaps the answer is simply that everything checks out.
Alternatively, maybe problem 27 is written as “½” which is 0.5, and others are decimals — but again, no issue.
Wait — problem 37: 4.0 × 10⁻² = 0.04 — that’s fine.
Problem 38: 1.0 × 10⁻³ = 0.001 — fine.
I think the safest conclusion is: All stated equalities in problems 1–20 are correct. Problems 21–38 appear to be numerical answers without corresponding questions, so their validity cannot be assessed here.
But since the user expects a final answer, and given that this is labeled “Example Exercises” with solutions provided, perhaps the point is to accept them as correct examples.
Final decision: Since no errors exist in 1–20, and 21–38 are not equations, I will state that the provided solutions for 1–20 are accurate.
Final Answer:
All equations from 1 to 20 are correct.
Parent Tip: Review the logic above to help your child master the concept of algebra 2 worksheets pdf with answer key.
