Laws of Logarithms – Definition and Examples

The laws of logarithms allow us to rewrite logarithmic expressions to form more convenient expressions. There are seven main laws of logarithms. These seven laws are useful for expanding logarithms, condensing logarithms, and solving logarithmic equations. The laws apply to logarithms of any base, but the same base must be used to apply each law.

Here, we will learn about the main laws of logarithms. In addition, we will look at several examples with answers to understand how to apply these laws to solve algebraic problems.

ALGEBRA
laws of logarithms

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Learning about the laws of logarithms with examples.

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ALGEBRA
laws of logarithms

Relevant for

Learning about the laws of logarithms with examples.

See examples

Descriptions of the laws of logarithms

Remember that a logarithm is the power to which a number must be raised to obtain another number. For example, the base 10 logarithm of 100 is 2, since 10 raised to the power of 2 equals 100:

$latex \log(100)=2$

because:

$latex {{10}^2}=100$

The base is the number that is being raised to a power. We can use logarithms with any base. If we wanted to, we could use the two as the base. For example, the logarithm with base two of eight is equal to three since two raised to the power of three is equal to eight:

$latex \log_{2}(8)=3$

given that

$latex {{2}^3}=8$

Now let’s look at the laws of logarithms.

Law 1: Product Law

This law tells us that the logarithm of a product is equal to the sum of the individual logarithms of the factors:

law of the logarithm of a product

Law 2: Quotient Law

This law tells us that the logarithm of the quotient of two quantities is equal to the logarithm of the numerator minus the logarithm of the denominator:

law of the logarithm of a quaotient

Law 3: Power Law

This law tells us that the logarithm of an exponential number is equal to the exponent multiplied by the logarithm of the base:

law of the logarithm of a power

Law 4: Law of Zero

If we have the logarithm of 1 where the base is $latex b> 0$, but $latex b \neq 0$, the result is equal to zero:

law of zero of logarithms

Law 5: Law of identity

The logarithm of the argument (inside the parentheses), where the argument is equal to the base is equal to 1. Since the argument is equal to the base, b must be greater than zero, but not can be equal to 1

law of identity of logarithms

Law 6: Law of the logarithm of the exponent

The logarithm of an exponential number, where its base is the same as the base of the logarithm is equal to the exponent:

Law of the logarithm of the exponent

Law 7: Law of the exponent of a logarithm

Raising the logarithm of a number to its base equals the number:

Law of the exponent of a logarithm

Laws of logarithms – Examples with answers

The following examples apply the laws of logarithms described above. Each example has its respective solution to look at the reasoning used and the laws of logarithms applied.

EXAMPLE 1

Evaluate the following expression: $latex \log_{2}(8)+\log_{2}(4)$.

We can apply the product rule to obtain:

$latex \log_{2}(8)+\log_{2}(4)=\log_{2}(8\times 4)$

$latex =\log_{2}(32)$

Now, we can write this expression in its exponential form to get the exponent:

$latex 32={{2}^5}$

Therefore, 5 is the correct answer.

EXAMPLE 2

Evaluate the expression: $latex \log_{3}(162)-\log_{3}(2)$.

Here, we have a subtraction of logarithms, so we can apply the quotient law:

$latex \log_{3}(162)-\log_{3}(2)=\log_{3}(\frac{162}{2})$

$latex =\log_{3}(81)$

Writing the argument in exponential form, we have:

$latex 81={{3}^4}$

Therefore, the answer is 4.

EXAMPLE 3

Find the value of $latex \log_{\sqrt{2}}(64)$.

We can use the power law twice to rewrite the expression and make it easier to solve the exercise:

$latex \log_{\sqrt{2}}(64)=\log_{\sqrt{2}}{{(2)}^6}$

$latex =6\log_{\sqrt{2}}(2)$

$latex =6\log_{\sqrt{2}}{{(\sqrt{2})}^2}$

$latex =6\times 2\log_{\sqrt{2}}(\sqrt{2})$

$latex =6\times 2(1)$

$latex =12$

EXAMPLE 4

Simplify the following expression $$\log_{5}(500)-2\log_{5}(2)+\log_{4}(32)+\log_{4}(8)$$

This expression seems a bit complicated at first glance, but we see that we have only logarithms with two different bases, 4 and 5.

Therefore, we can use the product rule to combine logarithms with base 4 and the quotient rule to combine the logarithms with base 5. However, first we have to apply the power law to the logarithm with base 5:

$$\log_{5}(500)-2\log_{5}(2)+\log_{4}(32)+\log_{4}(8)$$

$$=\log_{5}(500)-\log_{5}{{(2)}^2}+\log_{4}(32)+\log_{4}(8)$$

$$=\log_{5}(500)-\log_{5}(4)+\log_{4}(32)+\log_{4}(8)$$

$latex =\log_{5}(\frac{500}{4})+\log_{4}(32\times 8)$

$latex =\log_{5}(125)+\log_{4}(256)$

Now, we can rewrite the arguments as powers and apply the power law of logarithms:

$latex =\log_{5}({{5}^3})+\log_{4}({{4}^4})$

$latex =3\log_{5}(5)+4\log_{4}(4)$

$latex =3(1)+4(1)$

$latex =7$

EXAMPLE 5

Simplify the expression: $latex 2\log_{3}(5)+\log_{3}(40)-3\log_{3}(10)$.

We can start by using the power law to rewrite the individual logarithms. Then, we use the product law and the quotient law:

$latex 2\log_{3}(5)+\log_{3}(40)-3\log_{3}(10)$

$latex =\log_{3}({{5}^2})+\log_{3}(40)-\log_{3}({{10}^3})$

$latex =\log_{3}(25)+\log_{3}(40)-\log_{3}(1000)$

$latex =\log_{3}(\frac{25\times 40}{1000})$

$latex =\log_{3}(1)$

Finally, we apply the law of zero to solve:

$latex \log_{3}(1)=0$

EXAMPLE 6

Expand the logarithmic expression: $latex \log_{3}(27{{x}^2}{{y}^5})$.

We have a product of factors inside the parentheses, so we can apply the product law to write each factor separately. We can simplify individual logarithms when possible.

In this case we use the law of identity to simplify:

$$\log_{3}(27{{x}^2}{{y}^5})=\log_{3}(27)+\log_{3}({{x}^2})+\log_{3}({{y}^5})$$

$latex =\log_{3}({{3}^3})+\log_{3}({{x}^2})+\log_{3}({{y}^5})$

$latex =3\log_{3}(3)+2\log_{3}(x)+5\log_{3}(y)$

$latex =3(1)+2\log_{3}(x)+5\log_{3}(y)$

$latex =3+2\log_{3}(x)+5\log_{3}(y)$


Laws of logarithms – Practice problems

Use the laws of logarithms to solve the following problems. Choose an answer and check it to see that you selected the correct one. If you need help, you can look at the solved examples above or the list of the laws of logarithms.

Find the result of $latex \log_{10}(20)+\log_{10}(5)$.

Choose an answer






Find the result of $latex \log_{7}({{7}^3})$.

Choose an answer






Simplify the expression $latex {{16}^{\log_{4}(8)}}$.

Choose an answer






Simplify the expression $latex \log_{2\sqrt{2}}(64)$.

Choose an answer







See also

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