Interpreting Base-Ten Logarithms

By the end of this lesson, learners should be able to

1. Estimate a common logarithm from powers of ten.
2. Relate integer and non-integer exponents to number size.
3. Translate between notation forms accurately.

Key vocabulary

Estimate

A reasoned approximate value used before exact calculation.

Since 40 lies between 10 and 100, its log lies between one and two.

Interval

A range bounded by two numbers.

The logarithm of 250 lies in the interval two to three.

Characteristic

The integer part indicating a logarithm’s power-of-ten range.

A number from 100 to 999 has characteristic two.

Mantissa

The non-negative decimal part of a common logarithm.

A table can provide mantissa digits after locating the characteristic.

Lesson preview

Slide 1 · Where a logarithm sits between powersOpening · 18 min

Numbers between consecutive powers of ten have common logarithms between the corresponding integers. This gives a useful estimate before a table or calculator is used and links the size of a number to its logarithmic position. Use Figure 1 to see how numbers sit between consecutive powers of ten.

• Logarithms increase as positive numbers increase along the powers-of-ten ladder.
• Values between ten and one hundred have logarithms between one and two.
• A common logarithm represents a fractional exponent.
• Bounds provide a reasonableness check for answers.

Worked example. Compare 500 with 10² = 100 and 10³ = 1,000. Since 100 < 500 < 1,000, the log lies between 2 and 3. State the bound 2 < log₁₀(500) < 3. Answer: Between 2 and 3.

Check question: Between which two integers does ${\log }_{10}(0.4)$ lie?

1. $-2$ and $-1$
2. $-1$ and $0$
3. $0$ and $1$
4. $1$ and $2$

Show answer

$-1$ and $0$. Because $0.4$ is greater than $0.1$ (${10}^{-1}$) and less than $1$ (${10}^0$), its logarithm to base $10$ lies between $-1$ and $0$.

Slide 2 · Translate, bound, and checkPractice · 22 min

Teach the full lesson on the projector

This lesson, and every other Grade 1–10 CBE lesson, comes ready to project with the EAP ClassKit. It runs without electricity or internet.

• Both slides with diagrams and class activities
• Teacher notes, SNE adaptations and assessment checks
• Homework and the other 12 lessons in this sub-strand

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All 13 lessons in Indices and Logarithms

1. 1Reading Repeated Factors as IndicesExpress numbers in index form
2. 2Index Forms with a Specified BaseExpress numbers in index form
3. 3Discovering Product and Quotient LawsDerive the laws of indices using factors
4. 4Power of a Power and Identity CasesDerive the laws of indices using factors
5. 5Indices in Measurement and DesignApply the laws of indices in different situations
6. 6From Powers of Ten to Common LogsRelate index notation to logarithm notation to base 10
7. 7Interpreting Base-Ten LogarithmsRelate index notation to logarithm notation to base 10
8. 8Reading Common Logarithms in TablesDetermine common logarithms of numbers from mathematical tables and calculators
9. 9Comparing Table and Calculator LogarithmsDetermine common logarithms of numbers from mathematical tables and calculators
10. 10Logarithms for Multiplication and DivisionApply common logarithms in multiplication, division, powers and roots of numbers
11. 11Logarithms for Powers and RootsApply common logarithms in multiplication, division, powers and roots of numbers
12. 12Integrated Logarithmic ComputationApply common logarithms in multiplication, division, powers and roots of numbers
13. 13Why Indices and Logarithms MatterAppreciate the use of indices and common logarithms in mathematical computations