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From Zero to Engineer

Logarithms: definition, laws and change of base

What a logarithm is and how to read the notation log_a b, the eight laws of logarithms, the common and natural logarithm, and the formula that lets you compute a logarithm to any base with a calculator.

A logarithm is the inverse of raising to a power: instead of asking for the result, we ask for the exponent. If b equals a to the power c, then c is the logarithm of b to the base a. Everything else in this lesson follows from that single definition - the laws carry over what we already know about powers: multiplication turns into addition, division into subtraction, and raising to a power into multiplication.

Logarithms

b=ac,a>0,a≠1,b>0b = a^{c}, \quad a > 0, \quad a \neq 1, \quad b > 0
c=log⁡abc = \log_{a} b
c is the logarithm of b to the base a

a - the base

b - the argument (the number whose logarithm is being taken)

c - the result (the exponent)

A logarithm is the exponent to which a given base must be raised to obtain a given number.

a)x=log⁡216\text{a)}\quad x = \log_{2} 16
2x=162^{x} = 16
x=4x = 4
b)4=log⁡x81\text{b)}\quad 4 = \log_{x} 81
x4=81x^{4} = 81
x=3x = 3
c)2=log⁡7x\text{c)}\quad 2 = \log_{7} x
72=x7^{2} = x
x=49x = 49

The laws of logarithms

1)log⁡a(xy)=log⁡ax+log⁡ay\text{1)}\quad \log_{a}(xy) = \log_{a} x + \log_{a} y
the logarithm of a product is equal to the sum of the logarithms
2)log⁡a(x:y)=log⁡ax−log⁡ay\text{2)}\quad \log_{a}(x : y) = \log_{a} x - \log_{a} y
the logarithm of a quotient is equal to the difference of the logarithms
3)log⁡axn=n⋅log⁡ax\text{3)}\quad \log_{a} x^{n} = n \cdot \log_{a} x
the logarithm of a number raised to a power is equal to the exponent multiplied by the logarithm of the number
4)log⁡a1=0\text{4)}\quad \log_{a} 1 = 0
5)log⁡aa=1\text{5)}\quad \log_{a} a = 1
6)log⁡aax=x\text{6)}\quad \log_{a} a^{x} = x
7)alog⁡ax=x\text{7)}\quad a^{\log_{a} x} = x
8)1log⁡ba=log⁡ab\displaystyle \text{8)}\quad \frac{1}{\log_{b} a} = \log_{a} b

Worked examples

a)log⁡a(6.788⋅1.043)=log⁡a6.788+log⁡a1.043\text{a)}\quad \log_{a}(6.788 \cdot 1.043) = \log_{a} 6.788 + \log_{a} 1.043
b)log⁡a(19.112:0.054)=log⁡a19.112−log⁡a0.054\text{b)}\quad \log_{a}(19.112 : 0.054) = \log_{a} 19.112 - \log_{a} 0.054
c)log⁡a5.8891.2=1.2⋅log⁡a5.889\text{c)}\quad \log_{a} 5.889^{1.2} = 1.2 \cdot \log_{a} 5.889
d)log⁡81=0\text{d)}\quad \log_{8} 1 = 0
e)log⁡77=1\text{e)}\quad \log_{7} 7 = 1
f)log⁡327=3\text{f)}\quad \log_{3} 27 = 3
g)12log⁡124=4\text{g)}\quad 12^{\log_{12} 4} = 4
h)1log⁡34=log⁡43\displaystyle \text{h)}\quad \frac{1}{\log_{3} 4} = \log_{4} 3

Common logarithm and natural logarithm

The common logarithm is the logarithm to base 10, written simply as log a. The natural logarithm has base e and is written ln a.

The number e (Euler's number, Napier's constant, the base of the natural logarithm):

e≈2.718 28…e \approx 2.718\,28\ldots

Calculate the value of each of the following logarithms, correct to three decimal places (use a calculator).

a)log⁡5.321=0.726\text{a)}\quad \log 5.321 = 0.726
b)log⁡0.278=−0.556\text{b)}\quad \log 0.278 = -0.556
c)log⁡1=0\text{c)}\quad \log 1 = 0
d)log⁡(−1.005)\text{d)}\quad \log(-1.005)
it cannot be evaluated in the set of real numbers
e)ln⁡13.45=2.599\text{e)}\quad \ln 13.45 = 2.599
f)ln⁡0.278=−1.280\text{f)}\quad \ln 0.278 = -1.280
g)ln⁡0.00001=−11.513\text{g)}\quad \ln 0.00001 = -11.513
h)ln⁡(−0.001)\text{h)}\quad \ln(-0.001)
it cannot be evaluated in the set of real numbers

Change of base of a logarithm

log⁡ax=log⁡a10⋅log⁡10x\log_{a} x = \log_{a} 10 \cdot \log_{10} x
a)log⁡23.66=1.872\text{a)}\quad \log_{2} 3.66 = 1.872
log⁡210⋅log⁡103.66=3.322⋅0.563=1.872\log_{2} 10 \cdot \log_{10} 3.66 = 3.322 \cdot 0.563 = 1.872
b)log⁡3.40.293=−1.003\text{b)}\quad \log_{3.4} 0.293 = -1.003
log⁡3.410⋅log⁡100.293=1.882⋅(−0.533)=−1.003\log_{3.4} 10 \cdot \log_{10} 0.293 = 1.882 \cdot (-0.533) = -1.003
c)log⁡9.96.35=0.806\text{c)}\quad \log_{9.9} 6.35 = 0.806
log⁡9.910⋅log⁡106.35=1.004⋅0.803=0.806\log_{9.9} 10 \cdot \log_{10} 6.35 = 1.004 \cdot 0.803 = 0.806
d)log⁡7.347.34=1\text{d)}\quad \log_{7.34} 7.34 = 1

Frequently asked questions

What is a logarithm?

It is the exponent to which the base must be raised to obtain a given number. The notation log₂ 16 = 4 reads: two has to be raised to the fourth power to give 16.

Why can you not take the logarithm of a negative number?

Because the base is positive, and a positive number raised to any real power stays positive. So there is no exponent giving a negative result - hence the condition b > 0 in the definition.

What is the difference between log and ln?

The base. log is the common logarithm, to base 10, and ln is the natural logarithm, to base e, which is about 2.71828. Both have their own key on a calculator.

How do you compute a logarithm to a base your calculator does not have?

With the change of base formula: log_a x = log_a 10 · log₁₀ x. For log₂ 3.66 that gives 3.322 · 0.563, that is 1.872 - and all you need is the log key.

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Algebra: review exercises and the laws of exponents

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Logarithmic equations and rearranging formulas

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Logarithms: definition, laws and change of base | PhiBoard