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Intro to Logarithms

Learn what logarithms are and how to evaluate them. 

What you should be familiar with before taking this lesson

You should be familiar with exponents, preferably including negative exponents.

What you will learn in this lesson

You will learn what logarithms are, and evaluate some basic logarithms. This will prepare you for future work with logarithm expressions and functions.

What is a logarithm?

Logarithms are another way of thinking about exponents.
For example, we know that start color #11accd, 2, end color #11accd raised to the start color #0d923f, 4, end color #0d923f, start superscript, start text, t, h, end text, end superscript power equals start color #e07d10, 16, end color #e07d10. This is expressed by the exponential equation start color #11accd, 2, end color #11accd, start superscript, start color #0d923f, 4, end color #0d923f, end superscript, equals, start color #e07d10, 16, end color #e07d10.
Now, suppose someone asked us, "start color #11accd, 2, end color #11accd raised to which power equals start color #e07d10, 16, end color #e07d10?" The answer would be start color #0d923f, 4, end color #0d923f. This is expressed by the logarithmic equation log, start base, start color #11accd, 2, end color #11accd, end base, left parenthesis, start color #e07d10, 16, end color #e07d10, right parenthesis, equals, start color #0d923f, 4, end color #0d923f, read as "log base two of sixteen is four".
start color #11accd, 2, end color #11accd, start superscript, start color #0d923f, 4, end color #0d923f, end superscript, equals, start color #e07d10, 16, end color #e07d10, \Longleftrightarrow, log, start base, start color #11accd, 2, end color #11accd, end base, left parenthesis, start color #e07d10, 16, end color #e07d10, right parenthesis, equals, start color #0d923f, 4, end color #0d923f
Both equations describe the same relationship between the numbers start color #11accd, 2, end color #11accd, start color #0d923f, 4, end color #0d923f, and start color #e07d10, 16, end color #e07d10, where start color #11accd, 2, end color #11accd is the base and start color #0d923f, 4, end color #0d923f is the exponent.
The difference is that while the exponential form isolates the power, start color #e07d10, 16, end color #e07d10, the logarithmic form isolates the exponent, start color #1fab54, 4, end color #1fab54.
Here are more examples of equivalent logarithmic and exponential equations.
Logarithmic formExponential form
log, start base, start color #11accd, 2, end color #11accd, end base, left parenthesis, start color #e07d10, 8, end color #e07d10, right parenthesis, equals, start color #1fab54, 3, end color #1fab54\Longleftrightarrowstart color #11accd, 2, end color #11accd, start superscript, start color #1fab54, 3, end color #1fab54, end superscript, equals, start color #e07d10, 8, end color #e07d10
log, start base, start color #11accd, 3, end color #11accd, end base, left parenthesis, start color #e07d10, 81, end color #e07d10, right parenthesis, equals, start color #1fab54, 4, end color #1fab54\Longleftrightarrowstart color #11accd, 3, end color #11accd, start superscript, start color #1fab54, 4, end color #1fab54, end superscript, equals, start color #e07d10, 81, end color #e07d10
log, start base, start color #11accd, 5, end color #11accd, end base, left parenthesis, start color #e07d10, 25, end color #e07d10, right parenthesis, equals, start color #1fab54, 2, end color #1fab54\Longleftrightarrowstart color #11accd, 5, end color #11accd, start superscript, start color #1fab54, 2, end color #1fab54, end superscript, equals, start color #e07d10, 25, end color #e07d10

Definition of a logarithm

Generalizing the examples above leads us to the formal definition of a logarithm.
log, start base, start color #11accd, b, end color #11accd, end base, left parenthesis, start color #e07d10, a, end color #e07d10, right parenthesis, equals, start color #1fab54, c, end color #1fab54, \Longleftrightarrow, start color #11accd, b, end color #11accd, start superscript, start color #1fab54, c, end color #1fab54, end superscript, equals, start color #e07d10, a, end color #e07d10
Both equations describe the same relationship between start color #e07d10, a, end color #e07d10, start color #11accd, b, end color #11accd, and start color #0d923f, c, end color #0d923f:
  • start color #11accd, b, end color #11accd is the start color #11accd, start text, b, a, s, e, end text, end color #11accd,
  • start color #0d923f, c, end color #0d923f is the start color #0d923f, start text, e, x, p, o, n, e, n, t, end text, end color #0d923f, and
  • start color #e07d10, a, end color #e07d10 is called the start color #e07d10, start text, a, r, g, u, m, e, n, t, end text, end color #e07d10.

A helpful note

When rewriting an exponential equation in log form or a log equation in exponential form, it is helpful to remember that the base of the logarithm is the same as the base of the exponent.

Check your understanding

In the following problems, you will convert between exponential and logarithmic forms of equations.
Problem 1
Which of the following is equivalent to 2, start superscript, 5, end superscript, equals, 32?
Choose 1 answer:

Problem 2
Which of the following is equivalent to 5, cubed, equals, 125?
Choose 1 answer:

Problem 3
Write log, start base, 2, end base, left parenthesis, 64, right parenthesis, equals, 6 in exponential form.

Problem 4
4) Write log, start base, 4, end base, left parenthesis, 16, right parenthesis, equals, 2 in exponential form.

Evaluating logarithms

Great! Now that we understand the relationship between exponents and logarithms, let's see if we can evaluate logarithms.
For example, let's evaluate log, start base, 4, end base, left parenthesis, 64, right parenthesis.
Let's start by setting that expression equal to x.
log, start base, 4, end base, left parenthesis, 64, right parenthesis, equals, x
Writing this as an exponential equation gives us the following:
4, start superscript, x, end superscript, equals, 64
4 to what power is 64? Well, start color #11accd, 4, end color #11accd, start superscript, start color #1fab54, 3, end color #1fab54, end superscript, equals, start color #e07d10, 64, end color #e07d10 and so log, start base, start color #11accd, 4, end color #11accd, end base, left parenthesis, start color #e07d10, 64, end color #e07d10, right parenthesis, equals, start color #1fab54, 3, end color #1fab54.
As you become more practiced, you may find yourself condensing a few of these steps and evaluating log, start base, 4, end base, left parenthesis, 64, right parenthesis just by asking, "4 to what power is 64?"

Check your understanding

Remember, when evaluating log, start base, start color #11accd, b, end color #11accd, end base, left parenthesis, start color #e07d10, a, end color #e07d10, right parenthesis, you can ask: "start color #11accd, b, end color #11accd to what power is start color #e07d10, a, end color #e07d10?"
Problem 5
log, start base, 6, end base, left parenthesis, 36, right parenthesis, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Problem 6
log, start base, 3, end base, left parenthesis, 27, right parenthesis, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Problem 7
log, start base, 4, end base, left parenthesis, 4, right parenthesis, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Problem 8
log, start base, 5, end base, left parenthesis, 1, right parenthesis, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Challenge problem
log, start base, 3, end base, left parenthesis, start fraction, 1, divided by, 9, end fraction, right parenthesis, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Restrictions on the variables

log, start base, b, end base, left parenthesis, a, right parenthesis is defined when the base b is positive—and not equal to 1—and the argument a is positive. These restrictions are a result of the connection between logarithms and exponents.
RestrictionReasoning
b, is greater than, 0In an exponential function, the base b is always defined to be positive.
a, is greater than, 0log, start base, b, end base, left parenthesis, a, right parenthesis, equals, c means that b, start superscript, c, end superscript, equals, a. Because a positive number raised to any power is positive, meaning b, start superscript, c, end superscript, is greater than, 0, it follows that a, is greater than, 0.
b, does not equal, 1Suppose, for a moment, that b could be 1. Now consider the equation log, start base, 1, end base, left parenthesis, 3, right parenthesis, equals, x. The equivalent exponential form would be 1, start superscript, x, end superscript, equals, 3. But this can never be true since 1 to any power is always 1. So, it follows that b, does not equal, 1.

Special logarithms

While the base of a logarithm can have many different values, there are two bases that are used more often than others.
Specifically, most calculators have buttons for only these two types of logarithms. Let's check them out.

The common logarithm

The common logarithm is a logarithm whose base is 10 ("base-10 logarithm").
When writing these logarithms mathematically, we omit the base. It is understood to be 10.
log, start base, 10, end base, left parenthesis, x, right parenthesis, equals, log, left parenthesis, x, right parenthesis

The natural logarithm

The natural logarithm is a logarithm whose base is the number e ("base-e logarithm").
Instead of writing the base as e, we indicate the logarithm with natural log.
log, start base, e, end base, left parenthesis, x, right parenthesis, equals, natural log, left parenthesis, x, right parenthesis
This table summarizes what we need to know about these two special logarithms:
NameBaseRegular notationSpecial notation
Common logarithm10log, start base, 10, end base, left parenthesis, x, right parenthesislog, left parenthesis, x, right parenthesis
Natural logarithmelog, start base, e, end base, left parenthesis, x, right parenthesisnatural log, left parenthesis, x, right parenthesis
While the notation is different, the idea behind evaluating the logarithm is exactly the same!

Why are we studying logarithms?

As you just learned, logarithms reverse exponents. For this reason, they are very helpful for solving exponential equations.
For example the result for 2, start superscript, x, end superscript, equals, 5 can be given as a logarithm, x, equals, log, start base, 2, end base, left parenthesis, 5, right parenthesis. You will learn how to evaluate this logarithmic expression over the following lessons.
Logarithmic expressions and functions also turn out to be very interesting by themselves, and are actually very common in the world around us. For example, many physical phenomena are measured with logarithmic scales.

What's next?

Learn about the properties of logarithms that help us rewrite logarithmic expressions, and about the change of base rule that allows us to evaluate any logarithm we want using the calculator.

Want to join the conversation?

  • aqualine seed style avatar for user gene gagne
    I didn't get much from the explanation of the challenge problem or 3 (1/9) for the logarithm
    (61 votes)
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  • leafers tree style avatar for user Ian John P. Alberba
    How about log base-2 of -8=3? Is it therefore "defined"? I thought it that way because it can be written in exponential form as (-2)^3=-8 ..... Can we then say that base (b) and argument (a) can be negative {and must simultaneously be at the same time...right? }............ :) since a negative integer raised to an odd whole number is negative............

    P.S. [Sorry if I made a mistake hahahahha was just curious. Anyways, I would really appreciate an explanation. Thanks in advance :) ]
    (23 votes)
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  • duskpin ultimate style avatar for user Deng Leon
    How do you evaluate logarithms using a calculator?
    (8 votes)
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    • primosaur seed style avatar for user Ian Pulizzotto
      A scientific calculator generally always has an ln (natural logarithm, or log base e) key. From the change of base theorem, log base a of b = (ln b)/(ln a). For example, you can calculate log base 3 of 5 by calculating (ln 5)/(ln 3) which should give approximately 1.465. (Note that if your calculator also has a log key, another way to calculate log base 3 of 5 is to calculate (log 5)/(log 3). You should still get about 1.465.)
      Have a blessed, wonderful day!
      (9 votes)
  • blobby blue style avatar for user mehrin farjana
    why isn't log(-8) with a base of -2 equal to 3?
    (1 vote)
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    • leaf orange style avatar for user A/V
      Even though technically that is correct as an exponential, as a logarithm it is undefined. You cannot have a negative base in a logarithm, and here is why:
      Keep in mind that the log(x), with any base, the output will be a real number no matter what as long as the input is >=0.

      Let's have a hypothetical that f(x) = log-2(x) is a function. It's inverse is f(x) = (-2)^x . All integers will work fine, however, as a normal log can take in any real value and output any real value, let's put a fraction in the exponential.

      (-2)^(1/2) = sqrt(-2). This is not a real number. Again, another fraction:
      (-2)^(1/4) = cubert(-2). Again, not a real number.

      If we were to graph this on the real plane (xy plane), the function would not be continuous as there would be outputs in the imaginary plane. So since although negative log bases can have a value for some integers, due to this rule, it makes neg log bases impossible.

      hopefully that helps !
      (19 votes)
  • duskpin ultimate style avatar for user Tony Stark
    I didn't get the part where you said logarithms can't b can't be negative. i mean if we have negative integers with odd power then why cant it be written in logarithmic form ?
    And secondly you said b cant be 1 so whats wrong with log with base 1 and power (a) 1?
    (3 votes)
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    • mr pink red style avatar for user andrewp18
      Logarithms are undefined for base 1 because there exist no real power that we could raise one to that would give us a number other than 1. In other words:
      1ˣ = 1
      For all real 𝑥. We can never have 1ˣ = 2 or 1ˣ = 938 or 1ˣ = any number besides 1.
      If the base of the logarithm is negative, then the function is not continuous. For instance, sure the logarithm is defined for even and odd powers of negative numbers (though even powers are positive and the odd powers a negative and this is a wild jumping behavior that will continue for all integers). However, what about values between the integers? For instance, what if I asked you what power I needed to raise -2 to in order to get 1/2? The answer is a complex number, and it can only be found with some knowledge of trigonometry and the de'Moivre's theorem. In other words, there are gaps between the integer powers where the function is only defined in the nonreal numbers. The only places where it is defined (in the real numbers) is for integer powers, and plotting just those clearly don't give a continuous curve.
      (10 votes)
  • blobby green style avatar for user piyush.sahoo456
    what is the need of log ? like we already have exponents , so why log too ?
    (3 votes)
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    • male robot donald style avatar for user Venkata
      I gave this example elsewhere too, but I think it fits here too.

      Say you want to go shopping and you don't have a personal vehicle. What would you do? Well, you'd hire a cab/Uber and go to the supermarket to get your stuff. Now, how would you come back? Clearly, you'll need a hire another cab/Uber which will get you back home.

      Same logic here. Raising a number to an exponent is all well and good, but we need something to come back to the original number. Hence, logarithms came in the picture.
      (8 votes)
  • blobby green style avatar for user yeliseyk24
    I first didn't understand, but now I know about this. Thank you soo much.
    (5 votes)
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  • duskpin tree style avatar for user Lawhon, Gina
    So I'm understanding how to solve these basic logarithms where everything works out to be a whole number. However, how do you solve something when it does not work out evenly? It may be something that I just have to keep going through this course to figure out, but I'm curious. (Is it just something I just have to punch into the calculator?)
    (4 votes)
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  • piceratops seed style avatar for user radgunti
    How can I solve 9^((x/4)+1) + 3 = 28 * 3^(x/4). Can It be solved using logarithms.
    (4 votes)
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  • duskpin seed style avatar for user Trinity Burns
    How in the world do the negative exponents work? I understand the concept with positive numbers but not negative ones.
    (3 votes)
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    • male robot hal style avatar for user Wompy
      Negative exponents are a way of writing powers of fractions or decimals without using a fraction or decimal. They tell us how many times to divide the base number. To simplify an expression with a negative exponent, you just flip the base number and exponent to the bottom of a fraction with a 1 on top. This is because division is the inverse operation of multiplication 1.

      For example, 2^-3 is equal to 1/(2^3) which is equal to 1/8.
      (3 votes)