Log Rules Made Simple: Understanding the Laws of Logarithms

1. Logarithm rules

In this article, you will learn the rules of logarithms, also known as log rules. The operations involving logarithms can be summarized in seven rules. These rules unveil some nice properties of logarithms, such as their capacity to turn multiplication into addition and powers into products. Such properties are useful when dealing with very large or very small numbers.

Within this article, you will see information on:

What is a logarithm?
Log and exponent rules.
The log rules math.
Examples.
What are the seven rules of logarithms?
How do you use logs to find powers?
What is log 2 in exponential form?
And more.

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2. What is a logarithm?

The logarithm is the inverse of a power, which you can think of as the power that a given base needs to be raised to get a specific number. This operation is mathematically described as:

\log_{\,b}(x) = y \qquad b^{\,y} = x

where $b$ is the base. We can see how it works by answering the following question: to what power must 2 be raised to get 8? By following the previous definition, we see that

\begin{split} 2^3 &= 8 \\[.4em] \log_{\,2} (8) &= 3 \end{split}

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3. Log and exponent rules

We can realize from the definition of the logarithm that $\log$ is the inverse function of an exponential. The connection between these two functions is responsible for the math behind the rules of logarithms. In order to deeply explore such a connection, let's go back to the equation $x = b^{\,y}$ , where the base is a generic exponential.

Among all the possible values we can choose for the base $b$ , there is a special one broadly applied in science and denoted as $b = e = 2.718281$ . The number $e$ was introduced by Leonard Euler, who derived its value in 1731. You can find more details about it by accessing our e calculator 🇺🇸.

The logarithm written in this special base is called the natural logarithm and represented by $\ln = \log_{\,e}$ . Then, since $\ln$ and $e$ are inverse functions, we can set the following log and exponent rules:

e^{\ln(x)} = x \qquad \mathrm{and} \qquad \ln\left(e^{x}\right) = x

4. The log rules math

As we pointed out, there are seven main rules of logarithm. In this section, we will explore each one in detail.

1. The product rule

This rule states that the logarithm of the product is the sum of the logarithms of the factors. The formula behind it is given by:

\log_{\,b} (A \cdot B )= \log_{\,b} A + \log_{\,b} B

2. The quotient rule

In the case of a quotient, the resultant logarithm is the logarithm of the numerator minus the logarithm of the denominator. This rule can be written as:

\log_{\,b}\frac{A}{B} = \log_{\,b}A-\log_{\,b}B

3. The power rule

The logarithm of an exponential number is the exponent times the logarithm of the base. Such a rule is described by the formula:

\log_{\,b}\left(A^{k}\right) = k\cdot\log_{\,b}A

4. The zero rule

If the base $b$ is positive and different from $1$ , then the logarithm of $1$ to any base is equal to zero:

\log_{\,b}(1) = 0

5. The identity rule

If $b \ge 0$ , then the logarithm of the base is equals to $1$ , or in other words:

\log_{\,b}(b) = 1

6. The inverse property of the logarithm

The log of the base up to an exponent is equal to the exponent. The formula for this rule is:

\log_{\,b}\left(b^{k}\right) = k

7. The inverse property of the exponent

As we saw in the previous section, raising the base to the logarithm of a number is equivalent to that number. This rule can be written as:

b^{\log_{\,b}(k)} = k

Besides the well-known log and exponent rules, we also have the so-called change-of-base formula, which is described by the equation below:

\log_{\,b}(x) = \frac{\log_{\,c}(x)}{\log_{\,c}(b)}

Such a formula enables us to rewrite logs to a base that simplifies the problem. Moreover, it unveils that all the logarithms are proportional to each other, since:

\log_{\,b} (x) = k\,\log_{\,c} (x) \qquad k = \frac{1}{\log_{\,c}(b)}

We know that it was a lot of theory. So let's put the log rules into practice.

5. Examples

Let's apply the log rules math with some examples.

Simplify $\log_{\,2} (8 \cdot 4)$ :

We can easily make this calculation by applying the product rule:

\begin{split} \log_{\,2} (8 \cdot 4)& = \log_{\,2} 8 + \log_{\,2} 4 \\[1em] & = 3+2 \\[1em] & =5 \end{split}

Simplify $\log_{\,5} (125^{1/3})$ :

The final answer can be found by using the power rule:

\begin{split} \log_{\,5} (125^{1/3})& = \frac{1}{3}\cdot\log_{\,5} (125) \\[1em] & = \frac{1}{3}\cdot \log_{\,5} (5^{3})\\[1em] & =\frac{1}{3}\cdot 3 \\[1em] & = 1 \end{split}

6. FAQs

The seven logarithm rules can be summarized as:

Product rule;
Quotient rule;
Power rule;
Zero rule;
Identity rule;
Inverse property of the logarithm; and
Inverse property of the exponent.

These rules define the algebra behind log and exponents.

You can use a log to find powers by following the steps below:

Take the equation b^x = y.
Take the logarithm of both sides:

log_b(b^x ) = log_b(y)
Apply the power rule:

x log_b(b) = log_b(y)
Solve the last equation for x:

x = log_b(y)/ log_b(b)

By considering that:

log 2 = log₁₀ 2

which is known as the common logarithm of 2, then, the exponential form of this log is:

10^0.3010 = 2

Thus, the exponential form will depend on the base of your logarithm.

This article was written by João Rafael Lucio dos Santos and reviewed by Steven Wooding.