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Logarithms

A base-10 logarithm is the exponent of ten that gives a number; the natural logarithm uses base e.

Practice 5: Mathematical Routines

Question set for this topic

Part 1 · Hook

Why this matters

Earthquake scales, the loudness of sound in decibels and the brightness of stars all use logarithms, because each covers a range of millions or billions. A log turns "a thousand times bigger" into "3 more". Chemistry uses the same trick for quantities that change by powers of ten.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. What is 10⁻³ × 10⁵?

  1. 10²
  2. 10⁻¹⁵
  3. 10⁸
  4. 10⁻²
Show the answer

Multiplying powers adds the exponents: −3 + 5 = 2.

  • Correct: 10²:
  • 10⁻¹⁵:
  • 10⁸:
  • 10⁻²:

2. Which is 0.00047 in scientific notation?

  1. 4.7 × 10⁻⁴
  2. 4.7 × 10⁴
  3. 47 × 10⁻⁵
  4. 4.7 × 10⁻³
Show the answer

Move the point 4 places right to reach 4.7, so the exponent is −4.

  • Correct: 4.7 × 10⁻⁴:
  • 4.7 × 10⁴:
  • 47 × 10⁻⁵:
  • 4.7 × 10⁻³:

Part 4 · See it

See it first

A number line from 0.0001 to 10,000 in powers of ten, with logs −4 to 4 underneath: each factor of 10 adds 1 to the log, and numbers below 1 have negative logs.
A logarithm is the exponent: each step of 1 on the log scale is a factor of 10 in the number. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. A logarithm is the exponent that 10 must be raised tolog 1000 = 3 and log 0.001 = −3
  2. Multiplying numbers adds their exponentsthe log of a product is the sum of the logs, so a factor of 10 adds 1
  3. The natural log uses base e (about 2.718) instead of 10ln x = 2.303 log x, and mixing them up gives answers 2.303 times off
  4. An antilog raises the base to the log10ˣ undoes log and eˣ undoes ln

Part 6 · Key ideas

Key ideas

  • log x (base 10) is the power of ten that gives x: log 10⁻⁵ = −5. Numbers between 0 and 1 have negative logs.
  • log (A × B) = log A + log B; log (A/B) = log A − log B; log (Aⁿ) = n log A.
  • ln x is the natural log, base e. ln x = 2.303 log x. Use the key the equation names.
  • Digits after the decimal point in a log are its significant figures: log (3.2 × 10⁻⁴) = −3.49. Undo log with 10ˣ and ln with eˣ (the antilog).

Part 7 · Misconception

A common mistake

The wrong idea: log and ln are two names for the same button, so either one works.

What actually happens: They use different bases. ln x is 2.303 times log x, so using the wrong one makes the answer 2.303 times off. Equations that say ln need LN; equations that say log need LOG.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

1. What is log 1000?

  1. 1000
  2. 0.001
  3. 3
  4. 30
Show the answer

A logarithm answers "10 to what power?". 10³ = 1000, so log 1000 = 3.

  • 1000: The log is the exponent, not the number itself.
  • 0.001: That is 10⁻³, the reciprocal of 1000.
  • Correct: 3: Right: 10³ = 1000, so the base-10 logarithm of 1000 is 3.
  • 30: log 1000 asks "10 to what power gives 1000?", which is 3.

2. Calculate log (3.2 × 10⁻⁴). Give the right number of decimal places.

Type a number.

Show the answer

log (3.2 × 10⁻⁴) = log 3.2 + log 10⁻⁴ = 0.505 − 4 = −3.49. The coefficient 3.2 has two significant figures, so the log has two decimal places.

  • Answer: -3.49

3. Calculate ln (0.50). Give the right number of decimal places.

Type a number.

Show the answer

ln (0.50) = −0.693, written −0.69: two significant figures in 0.50 give two decimal places.

  • Answer: -0.69

4. Find x if log x = 2.45. Give x with the right number of significant figures.

Type a number.

Show the answer

x = 10^2.45 = 281.8…, written 2.8 × 10². The two decimal places in 2.45 give two significant figures in x.

  • Answer: 2.8e+2

5. log 2 = 0.301. What is log 200?

  1. 0.602
  2. 60.2
  3. 2.301
  4. 200.301
Show the answer

Split 200 into 2 × 10². Then log 200 = 0.301 + 2 = 2.301.

  • 0.602: This is log 4 (2 × log 2). 200 is 2 × 100, not 2².
  • 60.2: This multiplies log 2 by 200. The log of a product adds, it does not multiply.
  • Correct: 2.301: Right: log (2 × 10²) = log 2 + log 10² = 0.301 + 2.
  • 200.301: Add log 100, which is 2, not 200.

6. A student needs log (4.7 × 10⁻³) but presses LN and gets −5.36. How far off is the answer, and what is the fix?

  1. It is correct, since ln and log are two names for one function
  2. It is 2.303 times too large in size; press LOG to get −2.33
  3. It is off by 10; divide −5.36 by 10
  4. It has the wrong sign; the answer is +5.36
Show the answer

log (4.7 × 10⁻³) = −2.33. The 2.303 factor between ln and log is a slip the exam readers see every year.

  • It is correct, since ln and log are two names for one function: They use different bases (e and 10) and give different numbers.
  • Correct: It is 2.303 times too large in size; press LOG to get −2.33: Right: ln x = 2.303 log x, so −5.36 ÷ 2.303 = −2.33.
  • It is off by 10; divide −5.36 by 10: The logs differ by a factor of 2.303 (ln 10), not 10.
  • It has the wrong sign; the answer is +5.36: Numbers below 1 have negative logs in both bases; the sign is right.

Part 9 · Summary

Summary

A base-10 logarithm is the exponent of ten that gives a number; the natural logarithm uses base e. Logs turn multiplying into adding, so a factor of ten adds 1 to log x. ln x = 2.303 log x. An antilog undoes a log: 10ˣ for log, eˣ for ln. The decimal places of a log match the significant figures of the number.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections