Use logarithms as inverse exponents, expand valid products and reject forbidden candidates.
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Try
A quantity doubles each period. Starting from 5, it reaches 40 after how many periods? The equation is 5·2^t=40, so 2^t=8 and t=3. A logarithm names this recovered exponent: log₂8=3.
Now ask when the same continuous model reaches 30. The equation becomes 2^t=6. Its exact solution is log₂6, even though that exponent is not an integer. If the model allows only whole periods, the first recorded amount reaching or exceeding 30 occurs at period 3; those are different questions.
Predict
Predict log₂(1/4) and log₂1 using inverse exponent reasoning. They are −2 and 0. Logarithms can have negative outputs even though their real inputs must be positive. Do not confuse a negative logarithm value with a logarithm of a negative number.
Experiment
Reverse an exponential table.
Exponent t
2^t
log₂(2^t)
−2
1/4
−2
−1
1/2
−1
0
1
0
1
2
1
3
8
3
Use the rows to test a product law: (1/2)×8=4, and the corresponding logarithms add to −1+3=2. Then test a proposed sum law with 1+1. log₂2=1, but log₂1+log₂1=0. The failed sum rule has a direct numerical counterexample.
Understand
For a base b>0 with b≠1, log_b u is the unique real exponent t satisfying b^t=u. Its real domain is u>0. The notation ln means logarithm with base e. Changing bases gives log_b u=ln u/ln b, with the same base and argument restrictions.
If u and v are positive, log_b(uv)=log_b u+log_b v and log_b(u/v)=log_b u−log_b v. Also log_b(u^r)=r log_b u for positive u and real r. These laws follow by writing u and v as powers of b and applying exponent rules. They do not authorize taking real logarithms of negative factors merely because their product is positive.
For log₂(x−1)+log₂(x+1)=3, first require x>1. Combining gives log₂(x²−1)=3, hence x²=9 and candidates ±3. Only 3 belongs to the original domain. The negative candidate makes both original arguments negative. Checking just the combined expression would incorrectly restore it.
The identity ln(x²)=2 ln x illustrates another domain distinction. Its left side exists for any nonzero real x; its right side only for positive x. A valid broader statement is ln(x²)=2 ln|x| for x≠0. The absolute value is essential, not decorative notation.
For an exponential equation 5^t=7, logarithms yield t=ln7/ln5 exactly. A calculator supplies an approximation, not a new exact symbolic identity. Likewise, an equation yielding √6 should retain that exact expression if requested. The practice uses written or genuine choice responses where the exact checker does not support symbolic roots.
Apply
Begin each logarithmic equation with a domain line. After using laws and solving, filter every candidate through the original individual arguments. In the written identity task, compare domains before manipulating expressions; an equality can be true on one domain and undefined outside it.
When interpreting a logarithm in context, state what exponent it recovers and which time or scaling unit that exponent uses. A bare number does not explain the model.
Continue
The final lesson returns to integer-indexed change. Arithmetic and geometric sequences share ideas with linear and exponential functions, but their finite sums require clear indexing and separate treatment of exceptional parameters.
Apply · explain · revise
Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
Evaluate log₂(1/8).
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Solve log₃(x−1)=2 over the reals.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Solve log₂(x)+log₂(x−2)=3 over the reals.
Integers, fractions and finite decimals are accepted. Separate values with commas; type none for no solutions. Expressions such as sqrt(2) are not parsed by this checker.
For positive u and v, which identity is always valid?
Explain why ln(x²)=2 ln x is valid only for x>0 as written. Give a real-domain identity valid for every nonzero real x and test x=−2.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Solve ln(x−1)+ln(x+1)=ln 5 over the reals, giving exact roots and explaining every rejection.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
