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Mastering
AMC 10/12
Handbook · Chapter 6

Logarithms

What a logarithm is and the formulas that make it work: product, quotient and power rules and the change of base.

Book pages 124–126 · 8 entries

6.1 Basic Definitions

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Definition 6.1.1 · Logarithm

A logarithm is the power to which a number must be raised in order to get some other number.

Logarithms are written as

a=log⁡bna = \log_b n

where bb is the base and nn is the number.

Basically, we are asking how many times we need to multiply the base to get the number nn: what power do we need to raise the base to, to get nn?

Learn it:Definition 41.1.1 · LogarithmSection 41.1 Basic Definitions

Book p. 124

Theorem 6.1.2 · Converting to Logarithm and Exponents

log⁡xy=a  ⟹  xa=y\log_x y = a \implies x^a = y

The arrow works both ways: xa=yx^a = y also means log⁡xy=a\log_x y = a.

Learn it:Theorem 41.1.2 · Converting to Logarithm and Exponents

Book p. 124Checked numerically

6.2 Logarithmic Formulas

Theorem 6.2.1 · Important Formulas

log⁡aar=r\log_a a^r = r
log⁡abc=log⁡ab+log⁡ac\log_a bc = \log_a b + \log_a c
log⁡abc=log⁡ab−log⁡ac\log_a \frac{b}{c} = \log_a b - \log_a c
log⁡abc=clog⁡ab\log_a b^c = c \log_a b
log⁡ab⋅log⁡bc=log⁡blog⁡a⋅log⁡clog⁡b=log⁡clog⁡a=log⁡ac\log_a b \cdot \log_b c = \frac{\log b}{\log a} \cdot \frac{\log c}{\log b} = \frac{\log c}{\log a} = \log_a c
log⁡ba=1log⁡ab\log_b a = \frac{1}{\log_a b}
log⁡ba=log⁡dalog⁡db\log_b a = \frac{\log_d a}{\log_d b}
Correction

The book prints the fifth line as log⁡blog⁡a⋅log⁡clog⁡b=log⁡clog⁡a\frac{\log_b}{\log_a} \cdot \frac{\log_c}{\log_b} = \frac{\log_c}{\log_a}, with the numbers missing; it means log⁡blog⁡a⋅log⁡clog⁡b=log⁡clog⁡a\frac{\log b}{\log a} \cdot \frac{\log c}{\log b} = \frac{\log c}{\log a} (logs in any one base).

Learn it:Theorem 41.2.1 · Important FormulasSection 41.2 Logarithmic Formulas

Book p. 125Checked numerically

Remark 6.2.2

This last formula is known as the Base Change Formula and is the most useful of them all. Often in logarithm problems you can just expand your expression using this formula and simplify it to get the answer.

Learn it:Remark 41.2.2

Book p. 125

Remark 6.2.3

These formulas are extremely important for working with logarithms and should definitely be memorized.

Learn it:Remark 41.2.3

Book p. 125

Remark 6.2.4

If you ever forget which way the signs in these logarithm rules go, just try a small example like log⁡10100+log⁡101000=log⁡10100,000\log_{10} 100 + \log_{10} 1000 = \log_{10} 100{,}000. From there you can figure out the sum-of-logarithms identity.

Learn it:Remark 41.2.4

Book p. 125Checked numerically

Theorem 6.2.5 · Advanced Formulas

log⁡aman=nm\log_{a^m} a^n = \frac{n}{m}
log⁡a(1b)=−log⁡ab\log_a \left(\frac{1}{b}\right) = -\log_a b
log⁡1ab=−log⁡ab\log_{\frac{1}{a}} b = -\log_a b

Learn it:Theorem 41.2.6 · Advanced Formulas

Book p. 126Checked numerically

Remark 6.2.6

These formulas are less important and aren't necessary for most logarithm problems, but they are still good to know.

Learn it:Remark 41.2.7

Book p. 126