Change of base for logarithms

About this lesson

Precalculus › Exponential and logarithmic functions

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Change of base for logarithms

Warm-up

Three quick questions from earlier lessons.

Discover

Most calculators only have ln⁡\ln and log⁡\log (base 10). How could you find log⁡210\log_2 10 with one?

Hint: if y=log⁡210y = \log_2 10, then 2y=102^y = 10. Take ln⁡\ln of both sides.

Key idea

Any logarithm can be written with natural logarithms (or with base 10).

log⁡ac=ln⁡cln⁡a=log⁡clog⁡a\log_a c = \dfrac{\ln c}{\ln a} = \dfrac{\log c}{\log a}

The base aa always goes in the denominator.

Worked example

Evaluate log⁡48\log_4 8 exactly.

log⁡48=ln⁡8ln⁡4\log_4 8 = \dfrac{\ln 8}{\ln 4}
Change of base.
=3ln⁡22ln⁡2= \dfrac{3\ln 2}{2\ln 2}
8 = 2³ and 4 = 2², then the power rule.
=32= \dfrac{3}{2}
ln 2 cancels.

Your turn, guided

Your turn: evaluate log⁡927\log_9 27. Fill in the missing line.

log⁡927=ln⁡27ln⁡9\log_9 27 = \dfrac{\ln 27}{\ln 9}
Change of base.
=3ln⁡32ln⁡3= \dfrac{3\ln 3}{2\ln 3}
27 = 3³ and 9 = 3².
=32= \dfrac{3}{2}
ln 3 cancels.

Practice ★

  1. Place log⁡26\log_{2} 6 on the number line.
  2. Place log⁡224\log_{2} 24 on the number line.
  3. Place log⁡221\log_{2} 21 on the number line.
  1. Complete the change of base: log⁡420=ln⁡20□\log_{ 4 } 20 = \dfrac{\ln 20 }{\square}. What goes in the box?
  2. Complete the change of base: log⁡824=ln⁡24□\log_{ 8 } 24 = \dfrac{\ln 24 }{\square}. What goes in the box?
  3. Complete the change of base: log⁡416=ln⁡16□\log_{ 4 } 16 = \dfrac{\ln 16 }{\square}. What goes in the box?
  4. Complete the change of base: log⁡74=ln⁡4□\log_{ 7 } 4 = \dfrac{\ln 4 }{\square}. What goes in the box?
  5. Complete the change of base: log⁡49=ln⁡9□\log_{ 4 } 9 = \dfrac{\ln 9 }{\square}. What goes in the box?

Watch out!

Practice ★★

  1. Write log⁡816\log_{ 8 } 16 as a fraction in lowest terms.
  2. Write log⁡82\log_{ 8 } 2 as a fraction in lowest terms.
  3. Write log⁡981\log_{ 9 } 81 as a fraction in lowest terms.
  4. Write log⁡273\log_{ 27 } 3 as a fraction in lowest terms.
  5. Write log⁡9243\log_{ 9 } 243 as a fraction in lowest terms.

Exit ticket

Three questions to check you have got it.

  1. Evaluate log⁡816\log_{ 8 } 16 exactly. Hint: write 8 and 16 as powers of the same number.
  2. Evaluate log⁡84\log_{ 8 } 4 exactly. Hint: write 8 and 4 as powers of the same number.
  3. Evaluate log⁡279\log_{ 27 } 9 exactly. Hint: write 27 and 9 as powers of the same number.

Summary card

Change of base

log⁡ac=ln⁡cln⁡a\log_a c = \dfrac{\ln c}{\ln a}: the base goes below.

Check: log⁡25\log_2 5 must be between 2 and 3, because 22=42^2 = 4 and 23=82^3 = 8.

End of the lesson

Next lesson: Solve equations of the form e^(kx) = a Back to the chapter

Answers: Change of base for logarithms

A

  1. 1+ln⁡(3)ln⁡(2)1 + \frac{\ln{\left(3 \right)}}{\ln{\left(2 \right)}}
  2. ln⁡(3)ln⁡(2)+3\frac{\ln{\left(3 \right)}}{\ln{\left(2 \right)}} + 3
  3. ln⁡(21)ln⁡(2)\frac{\ln{\left(21 \right)}}{\ln{\left(2 \right)}}

B

  1. ln⁡(4)\ln{\left(4 \right)}
  2. ln⁡(8)\ln{\left(8 \right)}
  3. ln⁡(4)\ln{\left(4 \right)}
  4. ln⁡(7)\ln{\left(7 \right)}
  5. ln⁡(4)\ln{\left(4 \right)}

C

  1. 43\frac{4}{3}
  2. 13\frac{1}{3}
  3. 22
  4. 13\frac{1}{3}
  5. 52\frac{5}{2}

D

  1. 43\frac{4}{3}
  2. 23\frac{2}{3}
  3. 23\frac{2}{3}

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