Book Isle logarithms: distinguish products, sums and real-domain conditions

Adding logarithms does not mean adding their arguments. Read original expressions in Book Isle, fix the base and real domain, and check products through exponentiation. Use reading positions, available notes or external paper. No automatic symbolic simplification or universal proof from one calculator result is assumed.

1. Preserve the base, brackets and conditions

Prepare an original exercise and record edition, chapter/page and a nearby phrase. Work with real logarithms, base two and positive arguments. The general base must be positive and unequal to one, with both arguments positive individually. Consult definitions for unstated conditions instead of accepting two factors solely because their product is positive. Keep logarithms distinct from roots and powers. Mark an unreadable base unresolved and verify manual transcription against the source rather than assuming zoom implies math recognition.

2. Evaluate two values through exponentiation

Before using a rule, note that two cubed is eight and two squared is four, giving logarithms three and two. Pair each argument with the exponent producing it, not with the base as the answer. Retain the same base throughout. These are invented arithmetic examples, not acoustics, chemistry or finance data. If an external calculator is used, identify it as checking these inputs; explain the exponent relationship independently rather than treating display precision as algebraic justification.

3. Map the product rule to thirty-two

The OpenStax logarithm properties give the product rule under the stated conditions. Eight times four is thirty-two, two to the fifth is thirty-two, and three plus two is five. Map argument multiplication to exponent addition, not multiplication of the logarithm values. Here log_2 and * are declared plain-text mathematical notation, not Book Isle commands. Keep the original expression and simplified result on separate lines so every number can be traced.

2^3 = 8; 2^2 = 4; 2^5 = 32
log_2(8*4) = log_2(8) + log_2(4) = 3 + 2 = 5

4. Detect the sum error by reversing the definition

Create a labeled wrong control replacing the product with eight plus four while retaining answer five. Twelve is not two to the fifth, so the rewrite fails. Since twelve lies between eight and sixteen, its base-two logarithm lies between three and four, not five. Keep a bound instead of hiding the error behind decimals. Preserve the original product example. A sum inside an argument needs its own treatment; a coincidental equality at another input would not establish a general sum rule.

5. A positive product does not validate both arguments

Minus two times minus eight is sixteen, so a logarithm of the positive product can be considered, but splitting it into real logarithms of negative factors violates this domain. Check the original, each separated term and the rule’s conditions independently. No complex branch is introduced. Bases one, zero or negative are not admitted by this real-logarithm definition. Leave missing conditions unresolved rather than silently taking absolute values and claiming the original arguments stayed unchanged, even to preserve a preferred result of five or four.

6. Keep common-base and changed-base work separate

For a base-ten control, reevaluate the terms instead of carrying over base-two values three and two. The product rule retains its conditional form, but values change. A change-of-base formula from a reference needs its own base, numerator and denominator record, not an unexplained mixed calculation. No in-app base-setting button is required. Label original and control separately so the mathematical mapping, rather than consistent colors, explains the work.

7. Troubleshoot subscripts, brackets and order

Check common bases, the plus sign’s scope and positive arguments before reverse exponentiation. Copied text may flatten subscripts; retain declared notation without claiming a repaired native renderer. Retrieve scans by chapter/phrase rather than reimporting. Open exported notes and inspect multiplication, brackets, negative-factor domain and prohibitions. Preserve correction reasons instead of replacing only the final answer with five. Unreadable bases remain unresolved.

8. Return to definitions and recheck three cases

Exit and return, covering the results. Rebuild the product result five, the sum’s failure to equal five and the negative-factor restriction. Save edition/location, base/domain, original expression, mapping, reverse checks and unknowns. Another reader should explain the condition behind each step rather than memorize “logs become addition.” Recheck changed bases or domains in later reading. Notes support retrieval, not applicability decisions or universal proof from three invented checks.

Related tutorials and sources

Logarithmic axes, Function composition.

Checked on November 12, 2026 against the official Book Isle App Store listing and Apple public lookup: Book Isle 1.1.4, requiring iOS / iPadOS 15.6 or later. Use existing reading, available notes or external records and verify controls in your installed version. Original samples and manual calculations are not new automatic analysis features. No physical-device, private-file or universal export-environment testing is claimed.