Logarithmic functions have several important algebraic properties. Logarithm rules (sometimes shortened to “log rules”) describe how to perform algebraic manipulations with logarithmic functions. In this article, we’ll introduce these rules, examine why they are true, and walk through some examples of their uses.
To understand log rules, we first have to remember what logarithms are.
Definition of a Logarithm
By definition, the logarithm is the inverse function of exponentiation. Suppose t and b are positive numbers with the additional condition b ≠ 1. The logarithm base b of t, denoted x = log_b t , is defined to be the real number x such that b^x = t .
The base b = 1 is excluded because 1x = 1 for any real number x. It is also evident that the domain of any logarithmic function f(t) = logbt is any non-negative real number.
Here are a few examples on how to evaluate logarithms using this definition:
3^4 = 81 \qquad \Rightarrow \qquad log_3 81 = 4
2^{-3} = \dfrac{1}{8} \qquad \Rightarrow \qquad log_2 \dfrac{1}{8} = -3
10^2 = 100 \qquad \Rightarrow \qquad log_{10} 100 = 2
From the definition, we can also derive the following important identities:
log_b 1 = 0 ,, \quad \text{since} \quad b^0=1
log_b b = 1 ,, \quad \text{since} \quad b^1=b
Because logb x is the inverse function of bx, flipping the graph of y = bx across the line y = x gives the graph of y = logb x, which is illustrated in the image below:
Image Source: Wikimedia Commons
If b > 1, then logb x is an increasing function, because bx is an increasing function.
The Most Commonly Used Values for the Base
When we apply logarithms, the most commonly used values for the base are 10, 2, and the constant e, which is approximately equal to 2.71828. The use of these logarithms is so frequent that they have special names.
The Common Logarithm
Typically, we call the logarithm with base 10 the common logarithm. Usually, logarithms base 10 are written simply as log (without the base). In other words, if no base is displayed, then the base is assumed to be 10:
log x = log_{10} x
The common logarithm has many applications in science and engineering.
The Natural Logarithm
The logarithm with base e is called the natural logarithm. The natural logarithm is usually denoted as ln. Specifically,
ln ,x = log_{e} x , .
The natural logarithm appears frequently in mathematics and physics, because of its simpler derivative.
The Binary Logarithm
The function { lb }(x)=log _{ 2 } x is sometimes called the binary logarithm. Binary logarithms are less widespread relative to common and natural logarithms. They are generally used in computer science.
You can explore the graphs of the binary, common and natural logarithm functions on the image below:
Image Source: Wikimedia Commons
From this image, we can observe that the binary, common, and natural logarithm functions are all increasing functions. More generally, the function f(x) = logb x is an increasing function if b > 1, and a decreasing function if 0 < b < 1. We can also note that the function logb x diverges to infinity (gets bigger than any given positive number) if x grows to infinity, and b > 1. For b < 1, the function logb x also approaches infinity as x increases to infinity, but is negative in value (in other words, the function diverges to negative infinity). When x approaches zero, logb x approaches minus infinity if b > 1 and plus infinity if b < 1.
Also note that, because of the inverse properties of logarithms, the following equality holds:
log 10 = ln e = log_2 2 = 1 ,
The Inverse Properties of Logarithms
Since logarithms are the inverse functions of exponentiation, we can write
b^{log_b t} = t and log_b b^{x} = x .
Here, t and b are positive numbers, with b ≠ 1, and x is a real number. In other words, taking the xth power of b, then applying the base–b logarithm gives back x. Further, taking the base–b logarithm of t first then raising b to the power of that logarithm gives back t.
Here are a few examples of how the inverse properties can be used:
log_{10} 1000 = log_{10} 10^{3} = 3
log_{2} 32 = log_{2} 2^5 = 5
The Logarithm Product Rule
Among the other log rules, the logarithm product rule seems to be the most important. To derive this rule, we suppose x and y are positive numbers. Let log_b x = t and log_b y = a,, which means that x = b^t and y = b^a ,.
Then, we can write
log_b (x y) = log_b (b^t\cdot b^a) = log_b b^{(t+a)} = t + a
In other words, the logarithm of the product of variables x and y equals the sum of the logarithm of x and the logarithm of y with the same base. This relationship can be expressed as
log_b(x y)=log_b x + log_b y
This rule is known as the logarithm product rule. It allows us to split complicated logarithms into simpler sums:
log_2 6 = log_2 (2\cdot 3) = log_2 2 + log_2 3 = 1 + log_2 3
log (10x) = log 10 + log x = 1 + log x
The Logarithm Quotient Rule
We can derive the other important law, the quotient rule, analogously. The quotient rule states that the logarithm of a quotient equals the difference between the logarithm of the numerator and the logarithm of the denominator:
log_b \dfrac{x}{y}=log_b x - log_b y
This rule is especially useful for calculating the logarithms of fractions:
log 0.2 = log \dfrac{2}{10} = log 2 - log 10 = log 2 - 1
log 500 = log \dfrac{1000}{2} = log 1000 - log 2 = 3 - log 2
The Logarithm of a Multiplicative Inverse
As a special case of the logarithm quotient rule, take x = 1 in the formula for the logarithm of a quotient:
log_b \dfrac{1}{y} = log_b 1 - log_b y = - log_b y
The formula for the logarithm of a multiplicative inverse results in the following relations:
ln 0.1 = ln \dfrac{1}{10} = - ln 10
log 0.1 = log \dfrac{1}{10} = - log 10 = -1
The Logarithm Power Rule
We formulate the logarithm power rule as follows:
log_b (x^a) = alog_b x
In other words, the logarithm of an expression with an exponent is equivalent to the product of that exponent and the logarithm of the same expression without the exponent. We can derive this rule easily: Since
x^a = \left(b^{log_b x}\right)^a = b^{alog_b x} ,, log_b x^a = log_b \left(b^{alog_b x}\right) = alog_b x , .
If a is a natural number, we can understand the logarithm power rule as follows:
We can apply the logarithm power rule whenever x and b are positive numbers, b ≠ 1, and a is a real number. In particular, it can be used for fractional exponents. For example:
log \sqrt{10 x} = \dfrac 12 log(10 x) = \dfrac 12 (1+log x)
log \sqrt{0.07} = \dfrac 12 log 0.07 = \dfrac 12 log \dfrac{7}{100} = \dfrac 12 log 7 - 1
Change of Base for Logarithms
There are many instances where we need to know the value of a logarithm with an arbitrary base. For example, bases 2, 8, 16 are usually utilized in computer science, while the majority of calculators can only evaluate common and natural logarithms. We can resolve this problem by applying the logarithm base change rule.
If a, b, and y are positive numbers, with a ≠ 1 and b ≠ 1, we can derive the rule for changing the base of logarithms. Using the inverse properties of logarithms, we can write
y=b^{log_b y} = \left(a^{log_a b}\right)^{log_b y} = a^{log_a b ,\cdot, log_b y} ,
On the other hand, we have:
y = a^{log_a y} ,
By comparing the last two expressions, we arrive at the logarithm base change rule:
log_b y = \dfrac{log_a y}{log_a b} ,
A special case of this formula, suitable for use with calculators, is to take a = 10,or a = e, thus converting to common or natural logarithms in the following way:
log_b y = \dfrac{log y}{log b} = \dfrac{ln y}{ln b} ,
Remarkably, it doesn’t matter which standard base we use to calculate logby, as long as the same base is applied for both the numerator and denominator. For example, the expressions log2ex and log168 can be transformed into the following:
log_2 e^x = \dfrac{ln e^x}{ln 2} = \dfrac{x}{ln 2}
log_{16} 8 = \dfrac{log_2 8}{log_2 16} = \dfrac{3}{4}
Wrapping Everything Up
In this article, we have learned various rules for manipulating and simplifying expressions with logarithms. Log rules can be used to simplify expressions, to expand expressions, or to solve for values, depending on the given task. We hope this review will be helpful for you as you study algebra.
Let’s put everything into practice. Try this Algebra practice question:
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