Logarithm Base

Logarithm is an exponent that defines how many times a number can be multiplied to get another number. It is the power to which a number (the base) is raised to get another number. When talking about logarithms, there are terms you need to remember and be able to identify like the exponent and the base. 

Logarithm Base Logarithm Base

Create learning materials about Logarithm Base with our free learning app!

  • Instand access to millions of learning materials
  • Flashcards, notes, mock-exams and more
  • Everything you need to ace your exams
Create a free account
Contents
Table of contents

    We are going to focus more on the bases of logarithms. Hence, we should be able to identify a base when we see it. Let's get familiar with the basic formula associated with logarithms and then identify the base.

    by=X

    b is the base, y is the exponent to which the base is raised and X is the result obtained.

    It is also written as: logbX=y

    where log is short for logarithm.

    If you have 23=8, 2 is the base, 3 is the exponent and 8 is the result obtained. It can also be written as log2 8=3. If we are able to identify the base in a logarithmic expression, we can deduce the meaning of a base.

    Logarithm base meaning

    Logarithm base is the subscript of the logarithm symbol (log). You can say that it is the number that carries or raises the exponent depending on the form of the expression (by=X or logbX=y). Let's take some examples to strengthen our understanding on identifying a base.

    Identify the base in the following.

    1. log3 9=2
    2. 54=625

    Solution

    1. The base here is 3. It is the subscript of the logarithm symbol log.
    2. The base here is 5. It is the number that raises the exponent 4.

    The popular forms of logarithms are the common logarithm and natural logarithm. The common logarithm is in base 10 written as log10 or just log and the natural logarithm is in base e written as loge or ln. When solving common logarithms with base 10, it's best to use a calculator. The calculator has a log button that will give you the answer. You can try to do it without a calculator if the numbers are small and easy to calculate but if otherwise, reach out for your calculator.

    Give the answers to the following.

    1. log1000
    2. log20
    3. log8
    SolutionWe can see that the questions are all in base 10.a. You can get the answer to log1000 using your calculator. Just press the log button and type in 1000 you will get the answer.log 1000 = 3

    This means that if you multiplying 10 in three places gives 1000 that is 103.

    It is also possible to do this without a calculator because we can calculate that103 = 1000.b. log 20 = 1.3010c. log 8 = 0.9031

    For the natural logarithm in base e, the e is called Euler's number which is 2.71828. When you want to solve this, you use the ln button in your calculator to get the answer.

    Let's see some more examples.

    Give the answers to the following

    a. loge7.3

    b. ln25

    c. loge33.98

    Solution

    1. To get the answer to loge7.3, you'll need a calculator. You will need to press the ln button on the calculator and then 7.3. The answer will appear after that.

    loge 7.3 =1.9878

    b. Using a calculator,ln25 = 3.2188

    c. Using a calculator,loge33.98 =3.5257

    Aside from the common logarithm and natural logarithm with bases 10 and e, logarithms can also have any base. The base can be any number. For examplelog756,log24+log68andlog100 -log210 are logarithms with different bases.

    Solving logarithms with different bases

    When you have logarithms with different bases, it means that you have a logarithmic equation or expression where the bases are of different numbers. The way to go about this is to use a formula called the change of base formula. The aim here is to make the different bases equal. That way, you will be able to get a solution easily. Let's take a look at what the change of base formula looks like.

    logbx=logaxlogab

    The logarithmic rules we would normally use are the same rules for solving logarithmic base. Let's see some of those rules.

    • logbx+logby=logbx y
    • logbx-logby=logbxy
    • logbxn=nlogbx
    • logbx=logby x=y
    We need these formulas to help us because our calculators can only solve logarithms in base "10" and base "e". Let's see how the change of base formula is used in the following examples.

    Simplify y = log220

    Solution

    The first thing is to change the base using the change of base formula. You can change the base to any number including base 10 and the natural logarithm e. You just have to make sure that they are both the same base. Doing this we will have:

    log220=log1020log102

    We will use a calculator to solve the numerator and denominator to get:

    log1020log102=1.30100.3010=4.32

    Let's see more examples.

    Solve log3x = log94

    Solution

    You will notice that there are different bases involved so we will use the change of base formula. We can change both base to 3 or 9 and you will still arrive at the same answer. Remember the aim is just to make sure that both bases are equal.

    We will use the change of base formula on the right hand side. This means we are making the bases 3.

    log3x=log34log39log3x=log34log332

    There is a logarithm law in the form logbbn=n. We will apply this law to the denominator present and we will have:

    logbbn=nlog332=2

    We will put the result "2" in the equation and continue solving.

    log3x=log342log3x=12log34

    There is another logarithm law in the form logaxn=nlogax. If we apply this we will get:

    log3x=log3412log3x=log32

    Using the rule that if logbx=logby, then x=y, our final answer will be:

    x=2

    Solve log94+log3x=3

    Solution

    The first thing is to make the bases the same. We can choose to make them both 9 or 3. Either way, we will arrive at the same answer. Let's make them both 3.

    The change of base formula is: logbx=logaxlogab

    We will use the change of base formula on the first term of the expression and we will get:

    log34log39+log3x=3

    If you observe, you will see that you can simplify the denominator with a calculator or manually. You can tell that the result is 2 because 3 squared is 9. So, we will now have:

    log342+log3x=3

    Let's multiply each term by 2

    log34+2log3x=6

    We can use the power logarithmic rule on the second expression which is logbxn=nlogbx

    We will now have log34+log3x2=6

    We can use the addition rule here which is logbx+logby=logbxy

    Therefore log34x2=6

    We will now take the anti log to get

    4x2=36

    What we simplify did here was to raise the base 3 to the power of 6.

    The next and final step is to find x

    x2=364

    Take the square root of both sides

    x2=364 x 2 =272x=272=13.5

    Logarithms are sometimes expressed in a graphical form and the base of the logarithmic function can affect the outcome of the graph. What happens is that the bigger the base, the smaller the curve. In other words, the larger the base, the closer the curve gets to the y-axis.

    Let's take an example

    Plot the logarithmic expressions and observe the plot.

    y=log2x and y=log10x

    Solution

    What you need to do is make a table for both expressions and plot the graph.

    For y=log2x

    xy
    10
    21
    42

    For y=log10x

    xy
    10
    20.3
    40.6

    We will now plot the graph

    Bases of Logarithms Logarithm graphs StudySmarter

    You can see that y=log10x is closer to the y axis.

    Bases of Logarithms - Key takeaways

    • Logarithm base is either the subscript of the logarithm symbol (log) or the number that carries or raises the exponent depending on the from of the expression (by=X or logbX=y).
    • To solve logarithms with different bases, you use the change of base formula which is logbx=logaxlogab
    • The bigger the logarithm base, the smaller the curve on the graph. In other words, the larger the base, the closer the curve gets to the y-axis.
    Frequently Asked Questions about Logarithm Base

    How do you find the base of a logarithm?

    To find the base of a logarithm, you should be able to identify the base. The base is usually the subscript of the logarithm symbol (log). For example, the base in log4 is 2. 

    What is an example of a logarithm base function?

    An example of a logarithm base function is:

    y = log2 X.

    What are the four laws of logarithm?

    The four laws of logarithm are:

    • log(x) + log(y) = log(x y) (product law)
    •  log(x) - log(y) = log(x/y)  (division law)
    • log(xn) = n log(x)  (power law)
    • log(x) = loga (x) / loga (b)  (change of base law)


    How does the base of logarithm function affect a graph?

     The larger the base, the closer the curve gets to the y-axis. The larger the base, the smaller the curve.

    What is the meaning of logarithm base?

    Logarithm base is either the subscript of the logarithm symbol (log) or the number that carries or raises the exponent depending on the from of the expression (by = X or loga(X) =y).

    Discover learning materials with the free StudySmarter app

    Sign up for free
    1
    About StudySmarter

    StudySmarter is a globally recognized educational technology company, offering a holistic learning platform designed for students of all ages and educational levels. Our platform provides learning support for a wide range of subjects, including STEM, Social Sciences, and Languages and also helps students to successfully master various tests and exams worldwide, such as GCSE, A Level, SAT, ACT, Abitur, and more. We offer an extensive library of learning materials, including interactive flashcards, comprehensive textbook solutions, and detailed explanations. The cutting-edge technology and tools we provide help students create their own learning materials. StudySmarter’s content is not only expert-verified but also regularly updated to ensure accuracy and relevance.

    Learn more
    StudySmarter Editorial Team

    Team Math Teachers

    • 8 minutes reading time
    • Checked by StudySmarter Editorial Team
    Save Explanation

    Study anywhere. Anytime.Across all devices.

    Sign-up for free

    Sign up to highlight and take notes. It’s 100% free.

    Join over 22 million students in learning with our StudySmarter App

    The first learning app that truly has everything you need to ace your exams in one place

    • Flashcards & Quizzes
    • AI Study Assistant
    • Study Planner
    • Mock-Exams
    • Smart Note-Taking
    Join over 22 million students in learning with our StudySmarter App

    Get unlimited access with a free StudySmarter account.

    • Instant access to millions of learning materials.
    • Flashcards, notes, mock-exams, AI tools and more.
    • Everything you need to ace your exams.
    Second Popup Banner