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# Logarithmic Scale|Definition & Meaning

## Definition

A **nonlinear** scale that is **frequently utilized** in the **process** of assessing a **wide variety** of values is known as a **logarithmic scale.** Every **period** is **increased** by nearly the base of a **logarithm** rather than **growing** in **equal increments** as the **logarithm progresses.** The **scales** of **base** ten as **well** as base **e** are the **most common** ones to use.

A **technique** for **charting** and **evaluating** a wide range of values is the use of a logarithmic scale. **Log scales** rise by an **exponential** ratio as **opposed** to linear functions, which **increase** or decrease in equal steps. When **graphing logarithms,** analysts frequently employ a power of 10 or a base e scale, where the incremental rise or fall by a factor of a **logarithmic** base.

The **graph** of a **function expressed** as a positive **logarithm** is shown **below** in the **following** figure.

## How Exactly Does One Use a Logarithmic Scale?

You can **generate** a scale by **utilizing** integer **increments** with such a **linear function,** and each **increment** will reflect the same amount of **units** throughout the **entire** scale. The scale that displays when using a **logarithmic function** does so in **increments** that correspond to the powers by which a value is being **multiplied.** For **instance,** you might see a scale of one along the axis of a linear graph, with **increments** that either rise or drop by one point. This scale **represents** increments **depending** on the power of a **certain** number on a **logarithmic** graph.

### Appropriate Times To Employ Logarithmic Scales

When **working** with data that contains **extreme outliers,** such as very large or very small **numbers,** log scales become **indispensable.** When viewing data points that differ **significantly** from one another in terms of **percentage,** you can also employ a **logarithmic scale.** In general, you would select **logarithmic** scales if **indeed** the data in the **graph covers** a huge range, like an **exponentially** growing rate. This is **because** logarithmic scales are **linearly proportional** to **logarithms.** The following is a list of **professions** that **frequently** make use of log scales in **their** work:

- Actuarial
**science**entails the**calculation**of**costs**and**risks**by an actuary, who may also make use of a**logarithmic**scale while**determining**various**insurance metrics.** - Log
**scales**are**frequently**used in**nuclear**and**internal medicine**by related to**healthcare professionals**to**quantify**factors such as**pH values, radioactive decay, health changes,**and**bacterial proliferation.** - In the fields of
**statistics**and**analytics, logarithmic**scales are used when**dealing**with. **Interest**rates are**typically presented**in**logarithmic increments**by**financial**advisors so that**clients**may more easily**comprehend**the**growth**of an**asset**or the collective wealth of a group of people.**Geologists**once**measured earthquake**activity on the Richter scale, a**logarithmic application.**

## The Formula for the Logarithmic Scale

When **graphing** a wide **range** of **values** using a **log** scale, each interval expands or contracts exponentially. Here is the **formula** for **calculating logarithms** so that the scale can be **determined:**

**\[y = log_b x \]**

The **b-variable** in the **formula** indicates the **base number,** while the **y-variable denotes** the **exponents** to which the base number is raised. The x **variable** reflects the result of raising the base to the y power. To use a **base-10** scale, one **would substitute** 10 for the **b-variable:**

**\[y = log_{10} x \]**

## An Explanation of the Logarithmic Equation’s Use

**Graphing** right to left on a **logarithmic** scale **involves multiplication** or **division** by **powers** of 10 or **any other predetermined exponential** value. The **formula** and the **methods** below can be used to **calculate** the **logarithmic** scale of any given **collection** of **numbers:**

### Replace the Y-variable With Its Value

**Logarithms** are a **straightforward** method for **simplifying** the solution of **exponential** functions. Find out what the value of the y-variable is by **employing** a base of 10. You are able to **compute** the **function** in order to **determine** the **x-variable** if you use this **value** for y. Take, for instance, the case **where** you are graphing a **logarithmic function** and you have a value of **1,500** for the y-axis. You **would replace** this with the **following** in the formula:

**\[1500 = log_{10} x \]**

### Perform the Computation for the Logarithmic Function

You **need** to do some **algebraic** work to **calculate** the log function before you can figure out what the x **variable** is. You may solve the log **problem** using the **example** from before for a y-value of **1,500** by switching to an **exponential function** and **calculating** the **following:**

**10x** = **1,500**

The **value** of the exponent, which is **required** to **figure** out the scale for the **graph,** can be obtained by **finding** the x-value.

### Find the Value of X

It is **possible** to calculate the **change** in the value of **each interval** by **solving** for the x-variable. In the earlier illustration, the x-value of **3.18** is calculated by calculating **10x = 1,500**. The x-value grows by a factor of 10 for every **tenfold** increase in the y-value; for **example,** a y-value = **1,500 would** result in an **x-value** of **31.76.**

## Representation of Logarithmic Scale Graphically

The **graph** of a function that is **written** as a **negative logarithm** can be seen below in the **figure** that has been **provided** for your **reference.**

The **graph** of a **function** expressed as a **negative** and **positive logarithm** is depicted in the **figure** below.

## A Numerical Example of a Logarithm Scale

**Find** the **logarithm** value for x **when** x is **equal** to **20.** Also, **represent** the value with a **graph.**

### Solution

**Given that:**

**x = 20**

We **have** to find the **log value** of 20:

**log(20) = 1.3**

The **following** figure **represents** the **logarithmic** graph for the **given** value of x, **which** is 20.

*All images/mathematical drawings were created with GeoGebra.*