Logarithms for Powers and Roots

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

1. Apply the power law to logarithms.
2. Apply fractional factors for roots.
3. Check answers by raising or rooting the recovered value.

Key vocabulary

Power law

A rule multiplying a logarithm by an exponent.

For 7⁴, multiply log₁₀(7) by four.

Exponent

A number indicating repeated multiplication of a base.

The exponent four means four factors of seven.

Fractional exponent

An exponent written as a fraction for a root.

The square root of 49 is 49^(1/2).

Root index

The number indicating which root is required.

The index three identifies a cube root.

Scale

The relative size or order of a numerical quantity.

A fourth power should be much larger than its base when the base is greater than one.

Recover

Find the original quantity from a transformed value.

An antilogarithm recovers a power after logarithm work.

Radicand

The number inside a root symbol.

In √144, 144 is the radicand.

Quotient

A result obtained by dividing one number by another.

The root method uses a logarithm quotient.

Lesson preview

Slide 1 · Powers multiply logarithmsOpening · 10 min

For a positive number raised to a whole-number power, the logarithm of the result equals the exponent multiplied by the logarithm of the base. This converts repeated multiplication into a single multiplication of logarithm values. Use Figure 1 to visualise how the exponent transforms into a multiplier.

• The base number must be positive for real logs.
• Multiply the logarithm by the power exponent.
• Use an antilogarithm to recover the powered value.
• Estimate the scale before accepting the answer.

Worked example. Recognise four repeated factors of 9. Use the product law to add four logs. Compress the sum: log₁₀(9⁴) = 4log₁₀(9). Answer: 4log₁₀(9).

Check question: What is ${\log }_{10}(5^3)$ expressed in terms of ${\log }_{10}(5)$?

1. $3{\log }_{10}(5)$
2. $\frac{1}{3}{\log }_{10}(5)$
3. ${\log }_{10}(15)$
4. $5{\log }_{10}(3)$

Show answer

$3{\log }_{10}(5)$. The power law states that for a number raised to an exponent, the exponent becomes a multiplier for the logarithm. Three equal factors of $5$ produce three equal logarithm terms, giving $3{\log }_{10}(5)$.

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