Properties of Natural Logarithms

The four key properties of natural logarithms

Product property

The logarithm of a product property tells us that we can write the logarithm of a product as the sum of the individual logarithms of its factors:

Proof of this property

We can start with $latex x=\ln(p)$ and $latex y=\ln(q)$. If we now write these equations in their exponential form, we have:

⇒  $latex {{e}^x}=p$

⇒  $latex {{e}^y}=q$

We can multiply the exponential terms p and q to obtain:

$latex {{e}^x}\times {{e}^y}=pq$

We have powers with the same base, so we apply the product of exponents rule to add the exponents and combine the base:

$latex {{e}^{x+y}}=pq$

If we now take the natural logarithm to both sides, we have:

$latex \ln({{e}^{x+y}})=\ln(pq)$

Applying the property of the logarithm of a power (which we will see later), we have:

$latex (x+y)\ln(e)=\ln(pq)$

$latex (x+y)=\ln(pq)$

We can substitute the original values of x and y in the equation obtained:

$latex \ln(p)+\ln(q)=\ln(pq)$

Quotient property

The natural logarithms quotient property tells us that if we have a logarithm of a quotient, we can rewrite it as the logarithm of the numerator minus the logarithm of the denominator:

Proof of this property

We start with the equations $latex x=\ln(p)$ and $latex y=\ln(q)$. If we rewrite them in their exponential form, we have:

⇒  $latex {{e}^x}=p$

⇒  $latex {{e}^y}=q$

By dividing the exponential terms p and q, we have:

$latex \frac{{{e}^x}} {{{e}^y}}=\frac{p}{q}$

We can use the law of the quotient of exponents to simplify the expression on the left:

$latex {{e}^{x-y}}=\frac{p}{q}$

If we take the natural logarithm of both sides, we have:

$latex \ln({{e}^{x-y}})=\ln(\frac{p}{q})$

Applying the logarithm of a power rule (which we will see later), we have:

$latex (x-y)\ln(e)=\ln(\frac{p}{q})$

$latex (x-y)=\ln(\frac{p}{q})$

By substituting the original values of x and y in the equation obtained, we have:

$latex \ln(p)-\ln(q)=\ln(\frac{p}{q})$

Power property

The natural logarithm power property tells us that we can rewrite the logarithm of an exponential argument as follows:

Proof of this property

We start with $latex x=\ln(p)$ and we rewrite it in its exponential form:

⇒  $latex {{e}^x}=p$

If we raise to the power of n both sides of the equation, we have:

$latex {{({{e}^x})}^n}={{p}^n}$

⇒  $latex {{e}^{xn}}={{p}^n}$

Now, we take the natural logarithm of both sides:

$latex \ln({{e}^{xn}})=\ln({{p}^n})$

$latex xn~\ln(e)=\ln({{p}^n})$

$latex xn=\ln({{p}^n})$

Substituting the original value of x in the equation obtained, we have:

$latex n~\ln(p)=\ln({{p}^n})$

Reciprocal property

The natural logarithm of the reciprocal of x is the opposite of the natural logarithm of x:

Example:

$latex \ln(\frac{1}{3})=-\ln(3)$

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Other important properties of natural logarithms

In addition to the four properties of natural logarithms detailed above, there are other important properties of these logarithms that we need to know if we are studying natural logarithms. It is advisable to memorize these properties in order to simplify and solve logarithmic problems easily.

Natural logarithm of a negative number

The natural logarithm of any negative number is undefined.

Natural logarithm of zero

The natural logarithm of zero, that is, $latex \ln(0)$ is also undefined:

Natural logarithm of one

The natural logarithm of 1 is equal to zero:

Natural logarithm of infinity

The natural logarithm of infinity is equal to infinity:

Natural logarithm of e

The natural logarithm of the natural number, e, is equal to 1:

Property of the logarithm of a exponent

The logarithm of exponential e is equal to the exponent:

Property of the exponent of a logarithm

Raising e to the natural logarithm of a number equals the number:

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What is the difference between natural logarithms and other logarithms?

The main difference between natural logarithms and other logarithms is the base that is being used. Logarithms typically use a base of 10 (although it could be another value, which would be specified), while natural logarithms always use a base of e. This means that we can write:

$latex \ln(x)=\log_{e}(x)$

If you need to convert between logarithms and natural logarithms, you can use the following two equations:

$latex \log_{10}(x)=\frac{\ln (x)}{\ln (10)}$

$latex \ln(x)=\frac{\log_{10} (x)}{\log_{10} (e)}$

Apart from the difference in the base (which is a big difference), the laws of logarithms and the laws of natural logarithms are the same.

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See also

Interested in learning more about natural logarithms? Take a look at these pages:

• Laws of Natural Logarithms – Definition and Examples
• Change of Base of Logarithms – Rule and Examples