Properties of Logarithms

Introduction

• Logarithms are just another way of expressing exponents and can be used to solve problems that cannot be solved by the concept of exponents alone.

• In mathematics, logarithmic function properties are used to solve logarithmic problems.

• The division takes the final number and determines the count of the addition. Perhaps now you can appreciate how exponents and logarithms are a lot like multiplication and division.

• You will generally deal with a "base" in exponents and logarithms.

• The "base" of the exponent will be the same as the base of the logarithm.

• You have learned many basic mathematical properties such as commutative, and distribution that can be applied to algebra. In this tutorial, we will discuss the properties of logarithms.

Logarithms

• The logarithm is defined as a power that requires a number to be raised to get another value.

• Logarithms are just another way of expressing exponents and can be used to solve problems that cannot be solved by the concept of exponents alone

• Understanding the logs is not that difficult. To understand logarithms, it is enough to know that logarithmic equations are just another way to write exponential equations

• Logarithms and exponents are the opposite of each other. Logarithms were originally introduced by a mathematician named John Napier as an easy way to perform calculations. This concept was quickly adopted by other scientists, engineers, and others.

• The logarithm is defined as a power that requires a number to be raised to get another value.

• The most convenient way to represent large numbers. Logarithms have several important properties, proving that the multiplication and division of logarithms can also be described by logarithms of addition and subtraction

• $\mathrm{\log_{b}{a}\:=\:y}$, this is the logarithm of a concerning to base b. Here, y is the real number, and a and b are two positive real numbers, a is inside the log called argument, and b is at the bottom of the log called the base

Let us see the example

$\mathrm{2^{x}}$ is the same as taking x number of 2, and multiplying them together. Logarithm tells you how many times you have to multiply the base together to get the number.

For example −

$\mathrm{\log_{2}{x}}$ is counting how many 2 would need to be multiplied to make x. Using actual numbers might help.

$$\mathrm{2^{4}\:=\:16}$$

$$\mathrm{\log_{2}{16}\:=\:4}$$

These both deal with this series of multiplication − $\mathrm{2\times\:2\times\:2\times\:2}$

Logarithmic Rules

Log rules refer to logarithmic rules. These rules are derived from the rules of exponents, as logarithms are another way of describing exponents. The logarithmic formula is used to compress a set of logarithms into a single logarithm and expand the logarithms into a set of logarithms. Logarithmic formulas are the rules used to manipulate logarithms. Since logarithms is another way to describe exponents, we use exponent rules to derive the law of logarithms. There are mainly five rules that are primarily stated as follows −

Logarithmic properties

Product property

$$\mathrm{\log_{a}{xy}\:=\:\log_{a}{x}\:+\:\log_{a}{y}}$$

Quotient rule

$$\mathrm{\log_{a}{\frac{x}{y}}\:=\:\log_{a}{x}\:-\:\log_{a}{y}}$$

Power rule

$$\mathrm{\log_{a}{x}^{m}\:=\:m\log_{a}{x}}$$

Change of base rule

$$\mathrm{\log_{b}{c}\:=\:\frac{\log_{a}{c}}{\log_{a}{b}}}$$

Reciprocal rule

$$\mathrm{\log_{n}{m}\:=\:\frac{1}{\log_{m}{n}}}$$

The logarithmic formula is the same for all logarithms, including the natural logarithms. The natural logarithm is the logarithm with the base "e." It is represented by "ln." That is $\mathrm{\log_{e}\:=\:ln}$. That is, it does not write the base of the natural logarithm. If you look at "ln" automatically, you can see that its base is "e."

Properties of Logarithms

The logarithm is another way to describe an exponent. The logarithm is defined as a power that requires a number to be raised to get another value. The most convenient way to represent large numbers. Logarithms have several important properties, proving that the multiplication and division of logarithms can also be described by logarithms of addition and subtraction.

Logarithmic properties are

Product property

Suppose,

$$\mathrm{x\:=\:a^{n}\:and\:y\:=\:a^{m}}$$

Then the equivalent logarithmic forms are

$$\mathrm{\log_{a}{x}\:=\:n\:\:and\:\:\log_{a}{y}\:=\:m\:...............(1)}$$

Using the first rule of indices

$$\mathrm{xy\:=\:a^{n}\:\times\:a^{m}\:=\:a^{n\:+\:m}}$$

Now the logarithmic form of the statement $\mathrm{xy\:=\:a^{n\:+\:m}\:is\:\log_{a}{xy}\:=\:m\:+\:n}$

But $\mathrm{n\:=\:\log_{a}{x}\:and\:m\:=\:\log_{a}{x}}$ from (1) and so putting these results together we have

$$\mathrm{\log_{a}{xy}\:=\:\log_{a}{x}\:+\:\log_{a}{y}}$$

Quotient rule

As before, suppose

$$\mathrm{x\:=\:a^{n}\:and\:y\:=\:a^{m}}$$

with equivalent logarithmic forms

$$\mathrm{\log_{a}{x}\:=\:n\:and\:\log_{a}{y}\:=\:m\:\:.........{2}}$$

Consider $\mathrm{x\:\div\:y}$.

$$\mathrm{\frac{x}{y}\:=\:a^{n}\:\div\:a^{m}}$$

$$\mathrm{=\:a^{n\:-\:m}}$$

using the rules of indices.

In logarithmic form $\mathrm{\log_{a}\frac{x}{y}\:=\:n\:-\:m}$

which from (2) can be written

$\mathrm{\log_{a}\frac{x}{y}\:=\:\log_{a}x\:-\:\log_{a}y}$

This is the quotient law.

Change of base rule

$\mathrm{\log_{b}{c}\:=\:\frac{\log_{a}{c}}{\log_{a}{b}}}$

Reciprocal rule

$$\mathrm{\log_{n}{m}\:=\:\frac{1}{\log_{m}{n}}}$$

Uses of Logarithms

The logarithm is taught very early in one's mathematical career due to the enormous amount of application it has. I will list a few applications but keep in mind that there are so many more applications that depend on the context of the problem you are solving. Not only that, there is often a need for the log in the process of evaluating a limit, derivative, or even integral.

These are just a few of the applications for it, but this is what you will mostly use it for in application problems.

• pH levels in chemistry. The value of pH can be quite small, so the log is used, since it is base 10, to create a range for very small values.

• Interest in banking.

• Half- life of radioactive material.

• Deriving the charge of a capacitor.

• Range for loudness, this follows the same logic as with pH.

Solved Examples

Example1 − $\mathrm{x\:=\:\log_{2}{8}}$

Solution − $\mathrm{\:\:\:\:\:\:\:\:8\:=\:2^{x}}$

$\mathrm{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:8\:=\:2\:\times\:2\:\times\:2\:\times\:2}$

$\mathrm{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:8\:=\:2^{3}}$

$\mathrm{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:x\:=\:3}$

Example 2 − $\mathrm{\log_{8}{128}\:+\:\log_{8}{4}}$

Solution $\mathrm{\log_{8}{128}\:+\:\log_{8}{4}\:=\:\log_{8}{128\:\times\:4}}$

$\mathrm{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\log_{8}{512}}$

$\mathrm{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:Because\:512\:=\:8^{3}}$

$\mathrm{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\log_{8}{512}\:=\:3}$.

$\mathrm{\:\:\:\:\:\:\:\:\:\:So\:,\:\log_{8}{128}\:\log_{8}{4}\:=\:3}$.

Example 3 − $\mathrm{\log\:{x}\:=\:2\log{5}\:+\:\log{6}\:-\:\log{3}}$

Solution − $\mathrm{log\:x\:=\:\log5^{2}\:+\:\log6\:-\:\log3}$

$\mathrm{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:log\:x\:=\:\log(5^{2}\:\times\:6)\:-\:\log3}$

$\mathrm{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:log\:x\:=\:\log\:(\frac{5^{2}\:\times\:6}{3})\:=\:\log50}$

Example 4 − Write $\mathrm{\log_{7}{96}}$ as the sum of two logarithms.

Solution − $\mathrm{\log_{7}{96}\:=\:\log_{7}\:(3\:\times\:32)}$

$\mathrm{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\log_{7}{3}\:+\:\log_{7}{32}}$

Example 5 − Find x in $\mathrm{7^{x}\:=\:54}$

Solution − $\mathrm{x\:=\:\log_{7}{54}}$

$\mathrm{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:x\:=\:\frac{\log{54}}{\log{7}}}$

$\mathrm{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:x\:=\:2.0499}$

Conclusion

The log property is used to compress multiple logarithms terms into a single logarithm term or to expand a single logarithm term into multiple logarithms terms. Logarithm is different way to represent an exponent. Therefore, logarithmic properties are deduced from exponential properties.

Product property

$$\mathrm{\log_{a}{xy}\:=\:\log_{a}{x}\:+\:\log_{a}{y}}$$

Quotient rule

$$\mathrm{\log_{a}{\frac{x}{y}}\:=\:\log_{a}{x}\:-\:\log_{a}{y}}$$

Power rule

$$\mathrm{\log_{a}{x}^{m}\:=\:m\log_{a}{x}}$$

Change of base rule

$$\mathrm{\log_{b}{c}\:=\:\frac{\log_{a}{c}}{\log_{a}{b}}}$$

Reciprocal rule

$$\mathrm{\log_{n}{m}\:=\:\frac{1}{\log_{m}{n}}}$$

FAQs

1. What is the logarithm of 1?

The logarithm of 1 in any base is 0.

2. What is “e” in loge?

The symbol e is called the exponential constant and has a value approximately equal to 2.718.

3. What do you mean by logarithms?

The logarithm is defined as a power that requires a number to be raised to get another value.

4. What is the difference between log and ln?

log base ten is generally written as log, and log base e is written as ln.

5. Write the logarithmic properties.

• $\mathrm{\log_{a}{xy}\:=\:\log_{a}{x}\:+\:\log_{a}{y}}$

• $\mathrm{\log_{a}{\frac{x}{y}}\:=\:\log_{a}{x}\:-\:\log_{a}{y}}$

• $\mathrm{\log_{a}{x}^{m}\:=\:m\log_{a}{x}}$

• $\mathrm{\log_{b}{c}\:=\:\frac{\log_{a}{c}}{\log_{a}{b}}}$

• $\mathrm{\log_{n}{m}\:=\:\frac{1}{\log_{m}{n}}}$