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Amy Ferguson Moncure

IB AA ยท 12B Logarithms

Change of Base

Use log_b a = ln a / ln b. Simplify when both numbers share a prime base.

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Key Ideas IB wording Easy Medium Hard

\(\log_b(b^k)=k\)

\(\log A+\log B=\log(AB)\)

\(\ln a^n=n\ln a\)

\(\log_b a=\dfrac{\ln a}{\ln b}\)

What to Remember

Change of base

The formula is \(\log_b a=\dfrac{\ln a}{\ln b}\). You can also use \(\log\) in place of \(\ln\).

A matching power

\(\log_4 16=\dfrac{\ln 16}{\ln 4}=2\) because \(4^2=16\).

Same prime

\(\log_8 5=\dfrac{\log_2 5}{\log_2 8}=\dfrac{\log_2 5}{3}\).

Natural log in another base

\(\ln 4=\log_5 4\cdot\ln 5\), so \(\log_5 4=\dfrac{\ln 4}{\ln 5}\).

IB wording = log means base 10

When a question writes \(\log\) with no base, that means \(\log_{10}\). \(\ln\) means \(\log_e\), the natural logarithm.

A different base is written as a subscript, for example \(\log_2 8\).

Level 1

Easy

Write the log using natural logs.

Example
Write in terms of natural logarithms. \(\log_2 5\)

\(\log_b a=\dfrac{\ln a}{\ln b}\)

\(\log_2 5=\dfrac{\ln 5}{\ln 2}\)

Answer: \(\dfrac{\ln 5}{\ln 2}\)

1
Write in terms of natural logarithms. \(\log_3 7\)
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\(\log_3 7=\dfrac{\ln 7}{\ln 3}\)

Answer: \(\dfrac{\ln 7}{\ln 3}\)

2
Write in terms of natural logarithms. \(\log_5 2\)
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\(\log_5 2=\dfrac{\ln 2}{\ln 5}\)

Answer: \(\dfrac{\ln 2}{\ln 5}\)

3
Write in terms of natural logarithms. \(\log_6 6\)
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\(\log_6 6=1\)

Answer: \(1\)

4
Write in terms of natural logarithms. \(\log_4 4\)
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\(\log_4 4=1\)

Answer: \(1\)

5
Write in terms of natural logarithms. \(\log_2 8\)
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\(\log_2 8=3\)

Answer: \(3\)

Level 2

Medium

Change into the requested base.

Example
Write in base 2. \(\log_8 5\)

\(\log_8 5=\dfrac{\log_2 5}{\log_2 8}\)

\(\log_2 8=3\)

\(\log_8 5=\dfrac{\log_2 5}{3}\)

Answer: \(\dfrac{\log_2 5}{3}\)

1
Write in base 3. \(\log_9 7\)
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\(\log_9 7=\dfrac{\log_3 7}{\log_3 9}\)

\(\log_3 9=2\)

\(\log_9 7=\dfrac{\log_3 7}{2}\)

Answer: \(\dfrac{\log_3 7}{2}\)

2
Write in base 2. \(\log_{16} 5\)
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\(\log_{16} 5=\dfrac{\log_2 5}{\log_2 16}\)

\(\log_2 16=4\)

\(\log_{16} 5=\dfrac{\log_2 5}{4}\)

Answer: \(\dfrac{\log_2 5}{4}\)

3
Write in base 5. \(\ln 4\)
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\(\ln 4=\dfrac{\log_5 4}{\log_5 e}\)

\(\log_5 4=\dfrac{\ln 4}{\ln 5}\)

Answer: \(\dfrac{\ln 4}{\ln 5}\)

4
Write in base 4. \(\ln 7\)
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\(\log_4 7=\dfrac{\ln 7}{\ln 4}\)

Answer: \(\dfrac{\ln 7}{\ln 4}\)

5
Write in base 2. \(\log_4 32\)
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\(\log_4 32=\dfrac{\log_2 32}{\log_2 4}\)

\(\dfrac{5}{2}\)

Answer: 52

Level 3

Hard

Simplify fully when both numbers are powers of the same prime.

Example
Write in base 2 and simplify. \(\log_8 16\)

\(8=2^3\)

\(16=2^4\)

\(\log_8 16=\dfrac{4}{3}\)

Answer: 43

1
Write in base 3 and simplify. \(\log_9 27\)
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\(9=3^2\)

\(27=3^3\)

\(\log_9 27=\dfrac{3}{2}\)

Answer: 32

2
Write in base 2 and simplify. \(\log_8 32\)
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\(8=2^3\)

\(32=2^5\)

\(\log_8 32=\dfrac{5}{3}\)

Answer: 53

3
Write in base 5. \(\log_{25} 5\)
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\(\log_{25} 5=\dfrac{1}{2}\)

Answer: 12

4
Write in base 2. \(\log_{32} 4\)
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\(32=2^5\)

\(4=2^2\)

\(\log_{32} 4=\dfrac{2}{5}\)

Answer: 25

5
Write in base 3. \(\log_{27} 81\)
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\(27=3^3\)

\(81=3^4\)

\(\log_{27} 81=\dfrac{4}{3}\)

Answer: 43

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