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Amy Ferguson Moncure

IB AA · 12B Logarithms

Log Laws

Add, subtract, and pull out powers. Write condensed logs, or write in terms of p and q.

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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

Product and quotient

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

\(\log A-\log B=\log\dfrac{A}{B}\)

Power rule

\(\log(A^n)=n\log A\)

\(2\log a+3\log b=\log(a^2 b^3)\)

In terms of p and q

If \(p=\log_c a\) and \(q=\log_c b\), then \(\log_c(a^3/b)=3p-q\).

Same base

You can only add or subtract logs when the bases match. Keep the same base in the answer.

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 as a single logarithm.

Example
Write as a single logarithm. \(\log 3+\log 5\)

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

\(\log 3+\log 5=\log 15\)

Answer: \(\log 15\)

1
Write as a single logarithm. \(\log 20-\log 4\)
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\(\log A-\log B=\log\dfrac{A}{B}\)

\(\log 20-\log 4=\log 5\)

Answer: \(\log 5\)

2
Write as a single logarithm. \(2\ln a\)
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\(n\log A=\log(A^n)\)

\(2\ln a=\ln(a^2)\)

Answer: \(\ln(a^2)\)

3
Write as a single logarithm. \(\ln 7+\ln 2\)
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\(\ln 7+\ln 2=\ln 14\)

Answer: \(\ln 14\)

4
Write as a single logarithm. \(3\log_2 x\)
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\(3\log_2 x=\log_2(x^3)\)

Answer: \(\log_2(x^3)\)

5
Write as a single logarithm. \(\log_3 18-\log_3 2\)
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\(\log_3 18-\log_3 2=\log_3 9\)

\(\log_3 9=2\)

Answer: \(2\)

Level 2

Medium

p = log_c a and q = log_c b. Write the log in terms of p and q.

Example
Write in terms of p and q. \(\log_c\dfrac{a^3}{b}\)

\(p=\log_c a\)

\(q=\log_c b\)

\(\log_c\dfrac{a^3}{b}=3p-q\)

Answer: \(3p-q\)

1
Write in terms of p and q. \(\log_c(ab)\)
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\(\log_c(ab)=p+q\)

Answer: \(p+q\)

2
Write in terms of p and q. \(\log_c\dfrac{a}{b^2}\)
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\(\log_c\dfrac{a}{b^2}=p-2q\)

Answer: \(p-2q\)

3
Write in terms of p and q. \(\log_c(a^2 b^3)\)
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\(\log_c(a^2 b^3)=2p+3q\)

Answer: \(2p+3q\)

4
Write in terms of p and q. \(\log_c\sqrt{a}\)
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\(\log_c\sqrt{a}=\dfrac{1}{2}p\)

Answer: 12 p

5
Write in terms of p and q. \(\log_c\dfrac{\sqrt{b}}{a}\)
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\(\log_c\dfrac{\sqrt{b}}{a}=\dfrac{1}{2}q-p\)

Answer: 12 q − p

Level 3

Hard

Condense, then simplify. Watch coefficients in front of ln.

Example
Write as a single logarithm. \(2\ln a+6\ln b\)

\(2\ln a=\ln(a^2)\)

\(6\ln b=\ln(b^6)\)

\(\ln(a^2)+\ln(b^6)=\ln(a^2 b^6)\)

Answer: \(\ln(a^2 b^6)\)

1
Write as a single logarithm. \(3\ln x-\dfrac{1}{2}\ln y\)
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\(3\ln x=\ln(x^3)\)

\(\dfrac{1}{2}\ln y=\ln\sqrt{y}\)

\(\ln\dfrac{x^3}{\sqrt{y}}\)

Answer: \(\ln\dfrac{x^3}{\sqrt{y}}\)

2
Solve. \(\log_2 x+\log_2 5=4\)
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\(\log_2(5x)=4\)

\(5x=2^4\)

\(5x=16\)

\(x=\dfrac{16}{5}\)

Answer: 165

3
Solve. \(\log_3(x+4)+\log_3 2=2\)
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\(\log_3\bigl(2(x+4)\bigr)=2\)

\(2(x+4)=9\)

\(x+4=\dfrac{9}{2}\)

\(x=\dfrac{1}{2}\)

Answer: 12

4
Solve. \(\log_2(x+5)-\log_2(x-1)=3\)
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\(\log_2\dfrac{x+5}{x-1}=3\)

\(\dfrac{x+5}{x-1}=8\)

\(x+5=8x-8\)

\(13=7x\)

\(x=\dfrac{13}{7}\)

Answer: 137

5
Solve. \(\ln(x+2)-\ln(x-4)=2\)
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\(\ln\dfrac{x+2}{x-4}=2\)

\(\dfrac{x+2}{x-4}=e^2\)

\(x+2=e^2(x-4)\)

\(x-e^2 x=-4e^2-2\)

\(x=\dfrac{4e^2+2}{e^2-1}\)

Answer: \(\dfrac{4e^2+2}{e^2-1}\)

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