Lesson 4 of 5
Rewrite powers exactly and solve for an exponent without guessing.
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Worked example · 1 of 8
Worked example
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Solve . Check: , and the positive base 2 is one-to-one, so .
Big ideas
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For and , the exponential is one-to-one. Therefore means .
A number such as 16 can be written as , while a reciprocal such as is . Parentheses keep exponents on a whole rewritten power.
If , rewrite it as before solving . The base change contributes a factor to the exponent.
Substitute the resulting input into both original sides. This unit uses equations that have a useful exact common-base route; an arbitrary mismatch may need later methods.
Worked example
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Solve . Rewrite as , so and . Check .
Step-through
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Worked example: match powers of 3
Whole picture, 4 steps
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The fractional solution is allowed.
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Do this: Explain why the tempting shortcut fails.
Wrong idea: equate exponents before making the bases match. In the exponents are not x and 1 on a shared base. Rewrite as first.
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Flashcards
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32 is .
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Solve .
.
Rewrite as a power of 5.
.
Rewrite using base 2.
.
Why can exponents be equated for a common positive base other than 1?
The exponential is one-to-one.
Solve .
.
Solve .
.
Can equal −25 for a real x?
No.
Why does base 1 require care?
for every real x, so equal outputs do not force equal exponents.
True or false
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Self-test
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6 questions. One at a time.
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Self-test
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6 questions. One at a time.
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Write it
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Match
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