• Home
• /
• Blog
• /
• Difference Between Common Logarithm and Natural Logarithm

# Difference Between Common Logarithm and Natural Logarithm

July 14, 2022 In both exponential functions and logarithms, any number can be the base. However, there are two bases that are used so frequently that mathematicians have special names for their logarithms, and even scientific and graphing calculators include keys specifically for them! These are the common logarithm and natural logarithm.

The difference between $\text{log}$ and $\text{ln}$ is that $\text{log}$ is defined for base $10$ and $\text{ln}$ is denoted for base $e$. For example, $\text{log}$ of base $2$ is represented as $\text{log}_2$ and $\text{log}$ of base $e$, i.e. $\text{log}_e = \text{ln}$.

Let’s understand the difference between common logarithm and natural logarithm with examples.

## What is a Logarithm?

A logarithm is nothing but another way of expressing exponents. These are used to solve problems that cannot be solved using the concept of exponents only.

Logarithm and exponent are inverse forms of each other. For an expression $b^{x} = a$, the equivalent logarithmic expression will be $\log _b a = x$.

Here, “$\log$” stands for logarithm and is read as “Logarithm of a to the base b is equal to x“.

## What is a Common Logarithm?

The logarithm of a number to the base $10$ is known as a common logarithm. This system was first introduced by Henry Briggs. This type of logarithm is used for numerical calculations. The base $10$ in common logarithm is usually omitted while writing the expression.

For example, $x = \log_{10} 2$ is written as $x = \log 2$, which is same as $10^{x} = 2$.

## What is Natural Logarithm?

The logarithm of a number to the base $e$ (Euler’s constant) is known as a natural logarithm. This system was first introduced by John Napier and hence is also known as the Napierian logarithm. This type of logarithm is used for numerical calculations. If the base $e$ in the natural logarithm is omitted while writing the expression, it is written as $\ln x$.
For example, $x = \log_e 2$ is written as $x = \ln 2$, which is the same as $e^{x} = 2$.

Is your child struggling with Maths? We can help!
Country
• Afghanistan 93
• Albania 355
• Algeria 213
• American Samoa 1-684
• Andorra 376
• Angola 244
• Anguilla 1-264
• Antarctica 672
• Antigua & Barbuda 1-268
• Argentina 54
• Armenia 374
• Aruba 297
• Australia 61
• Austria 43
• Azerbaijan 994
• Bahamas 1-242
• Bahrain 973
• Belarus 375
• Belgium 32
• Belize 501
• Benin 229
• Bermuda 1-441
• Bhutan 975
• Bolivia 591
• Bosnia 387
• Botswana 267
• Bouvet Island 47
• Brazil 55
• British Indian Ocean Territory 246
• British Virgin Islands 1-284
• Brunei 673
• Bulgaria 359
• Burkina Faso 226
• Burundi 257
• Cambodia 855
• Cameroon 237
• Cape Verde 238
• Caribbean Netherlands 599
• Cayman Islands 1-345
• Central African Republic 236
• Chile 56
• China 86
• Christmas Island 61
• Cocos (Keeling) Islands 61
• Colombia 57
• Comoros 269
• Congo - Brazzaville 242
• Congo - Kinshasa 243
• Cook Islands 682
• Costa Rica 506
• Croatia 385
• Cuba 53
• Cyprus 357
• Czech Republic 420
• Denmark 45
• Djibouti 253
• Dominica 1-767
• Egypt 20
• Equatorial Guinea 240
• Eritrea 291
• Estonia 372
• Ethiopia 251
• Falkland Islands 500
• Faroe Islands 298
• Fiji 679
• Finland 358
• France 33
• French Guiana 594
• French Polynesia 689
• French Southern Territories 262
• Gabon 241
• Gambia 220
• Georgia 995
• Germany 49
• Ghana 233
• Gibraltar 350
• Greece 30
• Greenland 299
• Guam 1-671
• Guatemala 502
• Guernsey 44
• Guinea 224
• Guinea-Bissau 245
• Guyana 592
• Haiti 509
• Heard & McDonald Islands 672
• Honduras 504
• Hong Kong 852
• Hungary 36
• Iceland 354
• India 91
• Indonesia 62
• Iran 98
• Iraq 964
• Ireland 353
• Isle of Man 44
• Israel 972
• Italy 39
• Jamaica 1-876
• Japan 81
• Jersey 44
• Jordan 962
• Kazakhstan 7
• Kenya 254
• Kiribati 686
• Kuwait 965
• Kyrgyzstan 996
• Laos 856
• Latvia 371
• Lebanon 961
• Lesotho 266
• Liberia 231
• Libya 218
• Liechtenstein 423
• Lithuania 370
• Luxembourg 352
• Macau 853
• Macedonia 389
• Malawi 265
• Malaysia 60
• Maldives 960
• Mali 223
• Malta 356
• Marshall Islands 692
• Martinique 596
• Mauritania 222
• Mauritius 230
• Mayotte 262
• Mexico 52
• Micronesia 691
• Moldova 373
• Monaco 377
• Mongolia 976
• Montenegro 382
• Montserrat 1-664
• Morocco 212
• Mozambique 258
• Myanmar 95
• Namibia 264
• Nauru 674
• Nepal 977
• Netherlands 31
• New Caledonia 687
• New Zealand 64
• Nicaragua 505
• Niger 227
• Nigeria 234
• Niue 683
• Norfolk Island 672
• North Korea 850
• Northern Mariana Islands 1-670
• Norway 47
• Oman 968
• Pakistan 92
• Palau 680
• Palestine 970
• Panama 507
• Papua New Guinea 675
• Paraguay 595
• Peru 51
• Philippines 63
• Pitcairn Islands 870
• Poland 48
• Portugal 351
• Puerto Rico 1
• Qatar 974
• Romania 40
• Russia 7
• Rwanda 250
• Samoa 685
• San Marino 378
• Saudi Arabia 966
• Senegal 221
• Serbia 381 p
• Seychelles 248
• Sierra Leone 232
• Singapore 65
• Slovakia 421
• Slovenia 386
• Solomon Islands 677
• Somalia 252
• South Africa 27
• South Georgia & South Sandwich Islands 500
• South Korea 82
• South Sudan 211
• Spain 34
• Sri Lanka 94
• Sudan 249
• Suriname 597
• Svalbard & Jan Mayen 47
• Swaziland 268
• Sweden 46
• Switzerland 41
• Syria 963
• Sao Tome and Principe 239
• Taiwan 886
• Tajikistan 992
• Tanzania 255
• Thailand 66
• Timor-Leste 670
• Togo 228
• Tokelau 690
• Tonga 676
• Tunisia 216
• Turkey 90
• Turkmenistan 993
• Turks & Caicos Islands 1-649
• Tuvalu 688
• U.S. Outlying Islands
• U.S. Virgin Islands 1-340
• UK 44
• US 1
• Uganda 256
• Ukraine 380
• United Arab Emirates 971
• Uruguay 598
• Uzbekistan 998
• Vanuatu 678
• Vatican City 39-06
• Venezuela 58
• Vietnam 84
• Wallis & Futuna 681
• Western Sahara 212
• Yemen 967
• Zambia 260
• Zimbabwe 263
• Less Than 6 Years
• 6 To 10 Years
• 11 To 16 Years
• Greater Than 16 Years

## Examples

Ex 1: Which of the following statements is incorrect?

(a) $\text{log} \left(1 + 2 + 3 \right) = \text{log } 1 + \text{log } 2 + \text{log } 3$

(b) $\text{log } \left(2 + 3 \right) = \text{log }\left(2 \times 3 \right)$

(c) $\text{log}_{10} 10 = 1$

(d) $\text{log}_{10} 1 = 0$

The correct statement among the given statements is (b) $\text{log } \left(2 + 3 \right) = \text{log }\left(2 \times 3 \right)$.

Ex 2: Find the value of $\text{log } 9$, when $\text{log } 27$ equals $1.431$.

$\text{log } 27 = 1.431 => \text{log } 3^3 = 1.431 => 3 \times \text{log } 3 = 1.431 => \text{log } 3 = \frac{1.431}{3} => \text{log } 3 = 0.477$

Now, $\text{log } 9 = \text{log } 3^2 = 2 \times \text{log } 3 = 2 \times 0.477 = 0.954$

Ex 3: What is the value of $\text{log}_2 10$, when $\text{log}_{10}2 = 0.3010$?

$\text{log}_2 10 = \frac{1}{\text{log}_10 2} = \frac{1}{0.3010} =$

Ex 4: What is the Value of $\text{log}_{10} 80$, When $\text{log}_{10} 2 = 0.3010$?

$\text{log}_{10} 80 = \text{log}_{10} (8 \times 10) = \text{log}_{10} 8 + \text{log}_{10} 10$

$=\text{log}_{10} 2^3 + 1 = 3 times \text{log}_{10} 2 + 1 = 3 times 0.3010 + 1 = 0.903 + 1 = 1.903$

## Difference Between Common Logarithm and Natural Logarithm

These are the differences between common logarithms and natural logarithms.

## Applications of Logarithms

Some of the real-life applications of logarithms are:

• They are used for measuring the magnitude of earthquakes. $\log_{10} E = 11.8 + 1.5 M$ (where $\log$ refers to the logarithm to the base 1$0$, $E$ is the energy released in $ergs$ and $M$ the Richter magnitude).
• Logarithms are used for measuring the noise levels in $dBs$ (decibels). $L_p = 10 \times log_{10} \left(\frac{p}{p_{0}} \right) dB$, where $p$ = Measured Power, $p_0$ = Reference Power
• They are used to measure the $pH$ level of chemicals. $pH = log_{10}[H^{+}]$, where $H^{+}$ is concentration of molar hydrogen ion concentration
• Logarithms are used in radioactivity, mainly to detect the half-life of a radioactive element. $t_{\frac{1}{2}} = \frac {\ln \left(2 \right)}{c}$, $t_{\frac {1}{2}}$ is half life of decaying quantity and $c$ is decaying constant
• Logarithms are used to measure exponential growth or exponential decay. The best examples of these can be:
• The growth of money at a fixed rate of interest, say, for example, you have ₹$10,000$ in your bank account at an interest of $2\%$. With the help of logarithms, you’ll be able to know when your money’s going to reach ₹$12,000$.
• The growth of bacteria on a Petri dish. If you have a petri dish having bacteria taking up around $0.1\%$ space of the dish and you also know the fact that they divide every $30$ minutes, you’ll be able to calculate the time by which the bacteria will fill up that entire dish through the use of logarithms.
• Logarithms are used to measure radioactive decay in radiocarbon dating. Radiocarbon dating is the method of determining the age of an organic object by implementing the properties of radiocarbon (Carbon-$14$, $^{14}C$)
• Logarithms are used in specific calculations where multiplications are turned into additions.
• Logarithms are also implemented to calculate the exponential growth of the population.
• Logarithmic calculations also arise in calculus. Such calculations are used for several calculations in the real world.
• Logarithms are used in combinatorics problems. Combinatorics is a specific branch of mathematics that’s concerned with the study of finite discrete structures.

## Conclusion

Logarithms, both common and natural have immense applications in real-life problems. Logarithms help us to simplify the complex calculations that appear while solving problems.

## Practice Problems

1. Solve the equations
1. $2^{x} + 2 = 10$
2. $e^{2x} = 9$ (Take $\ln 3 = 1.0986$)
3. $e^{3x} = 8$ (Take $\ln 2 = 0.6931$)
2. Express $3^{x} \left(2^{2x} \right) = \frac{144}{25} \left( 5^{x} \right)$ in the form $a^{x} = b$. Hence, find $x$.
3. Solve without using a calculator: (Use $\log 2 = 0.3010$, $\log 3 = 0.4771$, $\ln 3 = 1.0986$, $\ln 5 = 1.6094$)
1. $\log3^{3}$
2. $\log 1$
3. $\log16^{2}$
4. $\ln e^{3}$
5. $\log 3$
6. $\log 3^{2}$
7. $\ln \sqrt {5}$
8. $\ln 729$
4. Solve, round to four decimal places.
1. $\log x = \log 2x^{2} – 2$
2. $\ln x + \ln \left( x + 1 \right) = 5$
5. How many digits are there in $264$ when $\text{log} 2 = 0.30103$?