Log Laws, Log Properties with Natural Logs

Addition and subtraction property

The addition property of logarithms relates the sum of two logarithms to the logarithm of a product.

Similarly, the subtraction property of logarithms relates the difference of two logarithms to the logarithm of a quotient.

We can prove these properties by using the corresponding properties of exponentials:

These two properties are especially valuable if we want to simplify expressions or solve equations involving logarithms.

Worked examples

example 1

Simplify the logarithmic expression $\ln12-\ln3$ln12−ln3.

Think: Since both logarithms are natural logarithms, they have the same base and so we can use the subtraction property of logarithms.

Do: To use the subtraction property of two logarithms, we can divide the arguments:

example 2

Rewrite $\ln5y$ln5y as the sum or difference of two logarithms.

Think: Since there is a product in the logarithm, we can use the addition property in reverse.

Do: So using the addition property, we can rewrite $\ln5y$ln5y in the form:

$\ln5+\ln y$ln5+lny

Practice questions

Question 1

Question 2

Power property

The power property of logarithms allows us to rearrange logarithmic expressions in unique ways. We can use this to simplify expression, or more easily compare two equations.

Exploration

Suppose we start with the quantity $\ln\left(x\right)$ln(x) and we triple it. We can write the resulting quantity as $3\ln\left(x\right)$3ln(x).

Next we can use repeated addition to write $3\ln\left(x\right)=\ln\left(x\right)+\ln\left(x\right)+\ln\left(x\right)$3ln(x)=ln(x)+ln(x)+ln(x). Now recall the following important property that is central to the idea of logarithms:

$\ln\left(a\right)+\ln\left(b\right)=\ln\left(ab\right)$ln(a)+ln(b)=ln(ab)

This says that the sum of the logarithm of two numbers is equal to the logarithm of the product of those two numbers. For the case of $3\ln\left(x\right)$3ln(x), we have

So we can see that $3\ln\left(x\right)=\ln\left(x^3\right)$3ln(x)=ln(x3). The same approach can be used for any other multiple of $\ln\left(x\right)$ln(x), and we can summarise this result like so.

Once we are familiar with the power property, we can bring in our knowledge of other index properties and apply them to expressions involving logarithms.

Specifically, consider the following properties:

• $\frac{1}{x^a}=x^{-a}$1xa​=x−a
• $\sqrt[b]{x^a}=x^{\frac{a}{b}}$^b√xa=xab​

When we take the logarithm of these expressions, we get:

• $\ln\left(\frac{1}{x^a}\right)=\ln\left(x^{-a}\right)=-a\ln(x)$ln(1xa​)=ln(x−a)=−aln(x)
• $\ln\left(\sqrt[b]{x^a}\right)=\ln\left(x^{\frac{a}{b}}\right)=\frac{a}{b}\ln(x)$ln(^b√xa)=ln(xab​)=ab​ln(x)

Let's look at an example that uses these properties.

Worked example 

Example 3

Write the expression $\ln\left(\frac{1}{x^4}\right)$ln(1x4​) as a multiple of $\ln\left(x\right)$ln(x).

Think: A multiple of $\ln\left(x\right)$ln(x) means an expression of the form $a\ln\left(x\right)$aln(x).

Do:

We have used the power property to write $\ln\left(\frac{1}{x^4}\right)$ln(1x4​) as $-4\ln\left(x\right)$−4ln(x), which is a multiple of $\ln\left(x\right)$ln(x).

Practice questions

Question 3

Question 4

Using multiple properties

We've already seen the addition, subtraction and power properties of logarithms. We are able to use these properties in order to simplify or rewrite natural logarithm expressions. The natural logarithm is just $\log_ex$loge​x represented as $\ln x$lnx.

The above properties can be used in order to simplify expressions, multiple properties may be used concurrently in order to simplify a single expression.

Worked examples

Example 4

Express $4\ln x+\ln y$4lnx+lny as a single logarithm.

Think: Which logarithm properties can we identify within the above expression? Since there is a sum of multiple logarithms we want to use the addition property, but we can only do this if the two logarithms have the same coefficient. Using the power property on $4\ln x$4lnx first will result in both of the logarithms having the same coefficients.

Do: First let's remove the coefficient of $\ln x$lnx using the power property:

Now let's express the sum of the two logarithms as a single logarithm using the addition property:

So, the simplest form of the sum of the two logarithms is $\ln\left(x^4y\right)$ln(x4y).

Example 5 

Express $\ln\left(\frac{x}{y^3}\right)$ln(xy3​) without using powers or fractions.

Think: Which logarithm properties can we identify within the above expression? Since there is a fraction within the logarithm we can use the subtraction property. The power of the variable $y$y means that we can also use the power property, but only once the single logarithm has been separated into two.

Do: First let's separate the logarithm into the difference of two logarithms using the subtraction property:

Now let's remove the power in the logarithm $\ln\left(y^3\right)$ln(y3) using the power property:

So, the expression represented without any fractions or powers is $\ln x-3\ln y$lnx−3lny.

Example 6

Simplify $\frac{2\ln\left(x^3\right)}{6\ln\left(x^2\right)}$2ln(x3)6ln(x2)​.

Think: Which logarithm properties can we identify in the above expression? The exponents suggest that we can use the power property on each logarithm, and then simplify the remaining expression.

Do: First let's remove the power of $x$x in $2\ln\left(x^3\right)$2ln(x3) using the power property:

Now let's remove the power in the logarithm in $6\ln\left(x^2\right)$6ln(x2) using the power property:

We can now simplify by cancelling a common factor of $6$6 and $\ln x$lnx from the numerator and denominator:

So the expression simplifies to $\frac{1}{2}$12​.

Practice questions

question 5

question 6

question 7