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Logarithms

Mathematics · FE Reference Handbook section

Mathematics
7 formulas
10 exam-style examples
~59 min
All Mathematics lectures

Handbook notes for this section

Definitions and conditions exactly as the handbook states them.

  • The logarithm of x to the Base b is defined by
  • To change from one Base to another:

Core formulas for this FE topic

Definitions, applicability, units, assumptions and worked examples for each relation.

Worked exam-style examples

The four ways this section is written on the real exam — thoughts first, then equations, then substitution.

Example 1
Change of logarithm base — Logarithms

Evaluate log₍5₎(164) using natural logarithms.

Given

  • Base=5Base = 5
  • Argument=164Argument = 164

Find

log₍5₎(164)

Start with the thinking

  • Calculators carry ln and log₁₀ only; convert with the change-of-base rule.

Step-by-step solution

  1. Change of base

    logb(x)=ln⁡x/ln⁡blog_b(x) = \ln x / \ln b

  2. Substituting

    log⁡(5)(164)=ln⁡(164)/ln⁡(5)=5.0999/1.6094\log ₍5₎(164) = \ln (164)/\ln (5) = 5.0999/1.6094

  3. Result — 3.1687

  4. Check

    53.169=164.0✓5^3.169 = 164.0 ✓
Answer:

3.169

Why the other options are there

  • 0.316 (ratio inverted)
  • 2.215 (base-10 log reported)

Reference: FE Reference Handbook — Mathematics → Logarithms

Example 2
Change of logarithm base — Logarithms (2)

Evaluate log₍5₎(719) using natural logarithms.

Given

  • Base=5Base = 5
  • Argument=719Argument = 719

Find

log₍5₎(719)

Start with the thinking

  • Calculators carry ln and log₁₀ only; convert with the change-of-base rule.

Step-by-step solution

  1. Change of base

    logb(x)=ln⁡x/ln⁡blog_b(x) = \ln x / \ln b

  2. Substituting

    log⁡(5)(719)=ln⁡(719)/ln⁡(5)=6.5779/1.6094\log ₍5₎(719) = \ln (719)/\ln (5) = 6.5779/1.6094

  3. Result — 4.0871

  4. Check

    54.087=719.0✓5^4.087 = 719.0 ✓
Answer:

4.087

Why the other options are there

  • 0.245 (ratio inverted)
  • 2.857 (base-10 log reported)

Reference: FE Reference Handbook — Mathematics → Logarithms

Example 3
Change of logarithm base — Logarithms (3)

Evaluate log₍5₎(786) using natural logarithms.

Given

  • Base=5Base = 5
  • Argument=786Argument = 786

Find

log₍5₎(786)

Start with the thinking

  • Calculators carry ln and log₁₀ only; convert with the change-of-base rule.

Step-by-step solution

  1. Change of base

    logb(x)=ln⁡x/ln⁡blog_b(x) = \ln x / \ln b

  2. Substituting

    log⁡(5)(786)=ln⁡(786)/ln⁡(5)=6.6670/1.6094\log ₍5₎(786) = \ln (786)/\ln (5) = 6.6670/1.6094

  3. Result — 4.1424

  4. Check

    54.142=786.0✓5^4.142 = 786.0 ✓
Answer:

4.142

Why the other options are there

  • 0.241 (ratio inverted)
  • 2.895 (base-10 log reported)

Reference: FE Reference Handbook — Mathematics → Logarithms

Example 4
Change of logarithm base — Logarithms (4)

Evaluate log₍2₎(306) using natural logarithms.

Given

  • Base=2Base = 2
  • Argument=306Argument = 306

Find

log₍2₎(306)

Start with the thinking

  • Calculators carry ln and log₁₀ only; convert with the change-of-base rule.

Step-by-step solution

  1. Change of base

    logb(x)=ln⁡x/ln⁡blog_b(x) = \ln x / \ln b

  2. Substituting

    log⁡(2)(306)=ln⁡(306)/ln⁡(2)=5.7236/0.6931\log ₍2₎(306) = \ln (306)/\ln (2) = 5.7236/0.6931

  3. Result — 8.2574

  4. Check

    28.257=306.0✓2^8.257 = 306.0 ✓
Answer:

8.257

Why the other options are there

  • 0.121 (ratio inverted)
  • 2.486 (base-10 log reported)

Reference: FE Reference Handbook — Mathematics → Logarithms

Example 5
Change of logarithm base — Logarithms (5)

Evaluate log₍2₎(256) using natural logarithms.

Given

  • Base=2Base = 2
  • Argument=256Argument = 256

Find

log₍2₎(256)

Start with the thinking

  • Calculators carry ln and log₁₀ only; convert with the change-of-base rule.

Step-by-step solution

  1. Change of base

    logb(x)=ln⁡x/ln⁡blog_b(x) = \ln x / \ln b

  2. Substituting

    log⁡(2)(256)=ln⁡(256)/ln⁡(2)=5.5452/0.6931\log ₍2₎(256) = \ln (256)/\ln (2) = 5.5452/0.6931

  3. Result — 8.0000

  4. Check

    28.000=256.0✓2^8.000 = 256.0 ✓
Answer:

8.000

Why the other options are there

  • 0.125 (ratio inverted)
  • 2.408 (base-10 log reported)

Reference: FE Reference Handbook — Mathematics → Logarithms

Example 6
Change of logarithm base — Logarithms (6)

Evaluate log₍10₎(691) using natural logarithms.

Given

  • Base=10Base = 10
  • Argument=691Argument = 691

Find

log₍10₎(691)

Start with the thinking

  • Calculators carry ln and log₁₀ only; convert with the change-of-base rule.

Step-by-step solution

  1. Change of base

    logb(x)=ln⁡x/ln⁡blog_b(x) = \ln x / \ln b

  2. Substituting

    log⁡(10)(691)=ln⁡(691)/ln⁡(10)=6.5381/2.3026\log ₍10₎(691) = \ln (691)/\ln (10) = 6.5381/2.3026

  3. Result — 2.8395

  4. Check

    102.839=691.0✓10^2.839 = 691.0 ✓
Answer:

2.839

Why the other options are there

  • 0.352 (ratio inverted)
  • 2.839 (base-10 log reported)

Reference: FE Reference Handbook — Mathematics → Logarithms

Example 7
Change of logarithm base — Logarithms (7)

Evaluate log₍5₎(412) using natural logarithms.

Given

  • Base=5Base = 5
  • Argument=412Argument = 412

Find

log₍5₎(412)

Start with the thinking

  • Calculators carry ln and log₁₀ only; convert with the change-of-base rule.

Step-by-step solution

  1. Change of base

    logb(x)=ln⁡x/ln⁡blog_b(x) = \ln x / \ln b

  2. Substituting

    log⁡(5)(412)=ln⁡(412)/ln⁡(5)=6.0210/1.6094\log ₍5₎(412) = \ln (412)/\ln (5) = 6.0210/1.6094

  3. Result — 3.7411

  4. Check

    53.741=412.0✓5^3.741 = 412.0 ✓
Answer:

3.741

Why the other options are there

  • 0.267 (ratio inverted)
  • 2.615 (base-10 log reported)

Reference: FE Reference Handbook — Mathematics → Logarithms

Example 8
Change of logarithm base — Logarithms (8)

Evaluate log₍5₎(826) using natural logarithms.

Given

  • Base=5Base = 5
  • Argument=826Argument = 826

Find

log₍5₎(826)

Start with the thinking

  • Calculators carry ln and log₁₀ only; convert with the change-of-base rule.

Step-by-step solution

  1. Change of base

    logb(x)=ln⁡x/ln⁡blog_b(x) = \ln x / \ln b

  2. Substituting

    log⁡(5)(826)=ln⁡(826)/ln⁡(5)=6.7166/1.6094\log ₍5₎(826) = \ln (826)/\ln (5) = 6.7166/1.6094

  3. Result — 4.1733

  4. Check

    54.173=826.0✓5^4.173 = 826.0 ✓
Answer:

4.173

Why the other options are there

  • 0.240 (ratio inverted)
  • 2.917 (base-10 log reported)

Reference: FE Reference Handbook — Mathematics → Logarithms

Example 9
Change of logarithm base — Logarithms (9)

Evaluate log₍2₎(287) using natural logarithms.

Given

  • Base=2Base = 2
  • Argument=287Argument = 287

Find

log₍2₎(287)

Start with the thinking

  • Calculators carry ln and log₁₀ only; convert with the change-of-base rule.

Step-by-step solution

  1. Change of base

    logb(x)=ln⁡x/ln⁡blog_b(x) = \ln x / \ln b

  2. Substituting

    log⁡(2)(287)=ln⁡(287)/ln⁡(2)=5.6595/0.6931\log ₍2₎(287) = \ln (287)/\ln (2) = 5.6595/0.6931

  3. Result — 8.1649

  4. Check

    28.165=287.0✓2^8.165 = 287.0 ✓
Answer:

8.165

Why the other options are there

  • 0.122 (ratio inverted)
  • 2.458 (base-10 log reported)

Reference: FE Reference Handbook — Mathematics → Logarithms

Example 10
Change of logarithm base — Logarithms (10)

Evaluate log₍2₎(607) using natural logarithms.

Given

  • Base=2Base = 2
  • Argument=607Argument = 607

Find

log₍2₎(607)

Start with the thinking

  • Calculators carry ln and log₁₀ only; convert with the change-of-base rule.

Step-by-step solution

  1. Change of base

    logb(x)=ln⁡x/ln⁡blog_b(x) = \ln x / \ln b

  2. Substituting

    log⁡(2)(607)=ln⁡(607)/ln⁡(2)=6.4085/0.6931\log ₍2₎(607) = \ln (607)/\ln (2) = 6.4085/0.6931

  3. Result — 9.2456

  4. Check

    29.246=607.0✓2^9.246 = 607.0 ✓
Answer:

9.246

Why the other options are there

  • 0.108 (ratio inverted)
  • 2.783 (base-10 log reported)

Reference: FE Reference Handbook — Mathematics → Logarithms

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