Why is the logarithmic property of equality true

Using the product rule for logarithms, we can rewrite this logarithm of a product as the sum of logarithms of its factors:. The product rule for logarithms can be used to simplify a logarithm of a product by rewriting it as a sum of individual logarithms. The final expansion looks like this. For quotients, we have a similar rule for logarithms. The quotient rule for logarithms says that the logarithm of a quotient is equal to a difference of logarithms.

Just as with the product rule, we can use the inverse property to derive the quotient rule. Factoring and canceling, we get. Next we apply the quotient rule by subtracting the logarithm of the denominator from the logarithm of the numerator. Then we apply the product rule. The quotient rule for logarithms can be used to simplify a logarithm or a quotient by rewriting it as the difference of individual logarithms.

Notice that the resulting terms are logarithms of products. To expand completely, we apply the product rule. There are exceptions to consider in this and later examples. Combining these conditions is beyond the scope of this section, and we will not consider them here or in subsequent exercises. One method is as follows:. Notice that we used the product rule for logarithms to find a solution for the example above. By doing so, we have derived the power rule for logarithms , which says that the log of a power is equal to the exponent times the log of the base.

Keep in mind that although the input to a logarithm may not be written as a power, we may be able to change it to a power. For example,. The power rule for logarithms can be used to simplify the logarithm of a power by rewriting it as the product of the exponent times the logarithm of the base. The argument is already written as a power, so we identify the exponent, 5, and the base, x , and rewrite the equivalent expression by multiplying the exponent times the logarithm of the base.

Next we identify the exponent, 2, and the base, 5, and rewrite the equivalent expression by multiplying the exponent times the logarithm of the base. Improve this page Learn More. Skip to main content. Module Exponential and Logarithmic Equations and Models.

Search for:. Properties of Logarithms Learning Outcomes Rewrite a logarithmic expression using the power rule, product rule, or quotient rule. Throughout your study of algebra, you have come across many properties—such as the commutative, associative, and distributive properties.

These properties help you take a complicated expression or equation and simplify it. The same is true with logarithms. There are a number of properties that will help you simplify complex logarithmic expressions. Since logarithms are so closely related to exponential expressions, it is not surprising that the properties of logarithms are very similar to the properties of exponents.

As a quick refresher, here are the exponent properties. Properties of Exponents. Product of powers:. Quotient of powers:. Power of a power:. This makes sense when you convert the statement to the equivalent exponential equation.

The result? Remember , so means and y must be 2, which means. Logarithm of a Product. Remember that the properties of exponents and logarithms are very similar. With exponents, to multiply two numbers with the same base , you add the exponents.

With logarithms, the logarithm of a product is the sum of the logarithms. The logarithm of a product is the sum of the logarithms:. Use the product property to rewrite. Use the product property to write as a sum. Simplify each addend, if possible. In this case, you can simplify both addends. Use whatever method makes sense to you. Another way to simplify would be to multiply 4 and 8 as a first step.

You get the same answer as in the example! In both cases, a product becomes a sum. Use the product property to rewrite log 3 9 x. In this case, you can simplify log 3 9 but not log 3 x.

If the product has many factors, you just add the individual logarithms:. Rewrite log 2 8 a, then simplify. The individual logarithms must be added, not multiplied. The logarithm of a product property says you separate the 8 and a into separate logarithms. Logarithm of a Quotient. You can use the similarity between the properties of exponents and logarithms to find the property for the logarithm of a quotient.

With exponents, to multiply two numbers with the same base, you add the exponents. To divide two numbers with the same base, you subtract the exponents. What do you think the property for the logarithm of a quotient will look like? As you may have suspected, the logarithm of a quotient is the difference of the logarithms.

With both properties: and , a quotient becomes a difference. Use the quotient property to rewrite. Use the quotient property to rewrite as a difference. However, the second expression can be simplified. What exponent on the base 2 gives a result of 2?

Which of these is equivalent to:. The individual logarithms must be subtracted, not divided. The correct answer is 4 — log 3 a. The logarithm of a quotient property says you separate the 81 and a into separate logarithms.

Logarithm of a Power. The remaining exponent property was power of a power:. The similarity with the logarithm of a power is a little harder to see. Use the power property to simplify log 3 9 4.