Common and Natural 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

Evaluate $y=log(321)$ to four decimal places 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?

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 $\ln(−500)$.