Common and Natural Logarithms

Sometimes we may see a logarithm written without a base. In this case, we assume that the base is 10. In other words, the expression $log(x)$ means $log_{10}(x)$. We call a base-10 logarithm a common logarithm. Common logarithms are used to measure the Richter Scale mentioned at the beginning of the section. Scales for measuring the brightness of stars and the pH of acids and bases also use common logarithms.

DEFINITION OF THE COMMON LOGARITHM

A common logarithm is a logarithm with base 10. We write $log_{10}(x)$ simply as $log(x)$. The common logarithm of a positive number $x$ satisfies the following definition.

For $x > 0$,

$y=log(x) \quad$ is equivalent to $10^y=x$

We read $log(x)$ as, "the logarithm with base $10$ of $x$ " or "log base $10$ of $x$".

The logarithm $y$ is the exponent to which $10$ must be raised to get $x$.

HOW TO

Given a common logarithm of the form $y=log(x)$, evaluate it mentally.

1. Rewrite the argument $x$ as a power of $10: 10^y=x$.
2. Use previous knowledge of powers of $10$ to identify $y$ by asking, "To what exponent must $10$ be raised in order to get $x$?"

EXAMPLE 5

Finding the Value of a Common Logarithm Mentally

Evaluate $y=log(1000)$ without using a calculator.

Solution

First we rewrite the logarithm in exponential form: $10^y=1000$. Next, we ask, "To what exponent must $10$ be raised in order to get $1000$?" We know

$10^3=1000$

Therefore, $log(1000)=3$.

TRY IT #5

Evaluate $y=log(1,000,000)$.

HOW TO

Given a common logarithm with the form $y=log(x)$, evaluate it using a calculator.

1. Press [LOG].
2. Enter the value given for $x$, followed by [ ) ].
3. Press [ENTER].

EXAMPLE 6

Finding the Value of a Common Logarithm Using a Calculator

Solution

• Press [LOG].
• Enter 321, followed by [ ) ].
• Press [ENTER].

Rounding to four decimal places, $log (321) \approx 2.5065$.

Analysis

Note that $10^2=100$ and that $10^3=1000$. Since 321 is between 100 and 1000, we know that $log(321)$ must be between $log(100)$ and $log(1000)$. This gives us the following:

$100 \quad < \quad 321 \quad < \quad 1000$

$2 \quad < \quad 2.5065 \quad < \quad 3$

TRY IT #6

Evaluate $y=log(123)$ to four decimal places using a calculator.

EXAMPLE 7

Rewriting and Solving a Real-World Exponential Model

The amount of energy released from one earthquake was 500 times greater than the amount of energy released from another. The equation $10^x=500$ represents this situation, where $x$ is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?

Solution

We begin by rewriting the exponential equation in logarithmic form.

$10^x \quad = 500$

$log (500) \quad = x$ Use the definition of the common log.

Next we evaluate the logarithm using a calculator:

• Press [LOG]
• Enter 500, followed by [ ) ].
• Press [ENTER].
• To the nearest thousandth, $log(500) \approx 2.699$.

The difference in magnitudes was about $2.699$.

TRY IT #7

The amount of energy released from one earthquake was 8,500 times greater than the amount of energy released from another. The equation \(10^x=8500 represents this situation, where $x$ is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?

Source: Rice University, https://openstax.org/books/college-algebra/pages/6-3-logarithmic-functions
This work is licensed under a Creative Commons Attribution 4.0 License.

The most frequently used base for logarithms is $e$. Base $e$ logarithms are important in calculus and some scientific applications; they are called natural logarithms. The base $e$ logarithm, $log_e(x)$, has its own notation, $\ln(x)$.

Most values of $\ln(x)$ can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base, $\ln1=0$. For other natural logarithms, we can use the $\ln$ key that can be found on most scientific calculators. We can also find the natural logarithm of any power of $e$ using the inverse property of logarithms.

DEFINITION OF THE NATURAL LOGARITHM

A natural logarithm is a logarithm with base $e$. We write $\log _{e}(x)$ simply as $\ln (x)$. The natural logarithm of a positive number $x$ satisfies the following definition.

For $x > 0$,

$y=\ln (x) \quad$ is equivalent to $e^{y}=x$

We read $\ln (x)$ as, "the logarithm with base $e$ of $x$" or "the natural logarithm of $x$"

The logarithm $y$ is the exponent to which e must be raised to get $x$.

Since the functions $y=e^{x}$ and $y=\ln (x)$ are inverse functions, $\ln \left(e^{x}\right)=x$ for all $x$ and $e^{\ln (x)}=x$ for $x > 0$.

HOW TO

Given a natural logarithm with the form $y= \ln(x)$, evaluate it using a calculator.

1. Press [LN].
2. Enter the value given for $x$, followed by [ ) ].
3. Press [ENTER].

EXAMPLE 8

Evaluating a Natural Logarithm Using a Calculator

Evaluate $y= \ln(500)$ to four decimal places using a calculator.

Solution

• Press [LN].
• Enter $500$, followed by [ ) ].
• Press [ENTER].

Rounding to four decimal places, $\ln(500) \approx 6.2146$

TRY IT #8

Evaluate $\ln(−500)$.