Use common 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 $\mathrm{log}\left(x\right)$ means ${\mathrm{log}}_{10}\left(x\right)$ . 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 is a logarithm with base $10$ . We write ${\mathrm{log}}_{10}\left(x\right)$ simply as $\mathrm{log}\left(x\right)$ . The common logarithm of a positive number $x$ satisfies the following definition.

For $x>0$ ,

$y=\mathrm{log}\left(x\right)\text{ is equivalent to }{10}^{y}=x$

We read $\mathrm{log}\left(x\right)$ 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 $x$ .

Given a common logarithm of the form $y=\mathrm{log}\left(x\right)$ , evaluate it mentally.

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

Finding the Value of a Common Logarithm Mentally

Evaluate $y=\mathrm{log}\left(1000\right)$ without using a calculator.

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, $\mathrm{log}\left(1000\right)=3$ .

Evaluate $y=\mathrm{log}\left(1,000,000\right)$ .

$\mathrm{log}\left(1,000,000\right)=6$

Given a common logarithm with the form $y=\mathrm{log}\left(x\right)$ , evaluate it using a calculator.

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

Finding the Value of a Common Logarithm Using a Calculator

Evaluate $y=\mathrm{log}\left(321\right)$ to four decimal places using a calculator.

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

Rounding to four decimal places, $\mathrm{log}\left(321\right)\approx 2.5065$ .

Analysis

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

$\begin{array}{ccccc}100& <& 321& <& 1000\\ 2& <& 2.5065& <& 3\end{array}$

Evaluate $y=\mathrm{log}\left(123\right)$ to four decimal places using a calculator.

$\mathrm{log}\left(123\right)\approx 2.0899$

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?

We begin by rewriting the exponential equation in logarithmic form.

$\begin{array}{lll}{10}^{x}\hfill & =500\hfill & \hfill \\ \mathrm{log}\left(500\right)\hfill & =x\hfill & \text{Use the definition of the common log}\text{.}\hfill \end{array}$

Next we evaluate the logarithm using a calculator:

Press [LOG].
Enter $500$ , followed by [ ) ].
Press [ENTER].
To the nearest thousandth, $\mathrm{log}\left(500\right)\approx 2.699$ .

The difference in magnitudes was about $2.699$ .

The amount of energy released from one earthquake was $\text{8,500}$ 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?

The difference in magnitudes was about $3.929$ .

Chapter 5.3: Logarithmic Functions

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