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Logarithms: FAQ

Frequently asked questions about logarithms

What are logarithms?

Logarithms are the inverse operation of exponentiation. We can use logarithms to find the exponent to which a given base must be raised in order to produce a particular result. For example, log28=3, because 23=8.

What is the constant e?

The constant e is a very important number in mathematics. It is an irrational number, which means it cannot be written exactly as a fraction or a decimal, but we often approximate it as e2.71828. It is the base of the natural logarithm, ln.
As n gets larger and larger, the sequence (1+1n)n gets closer and closer to e.
The constant e appears in various other contexts in mathematics and science, such as in statistics and in the study of exponential growth and decay.

What are some properties of logarithms?

There are a few key properties of logarithms that we use frequently:
  • logb1=0 for any base b
  • logbb=1 for any base b
  • logbxy=logbx+logby
  • logbxy=logbxlogby
  • logbxn=nlogbx

What is the change of base formula for logarithms?

The change of base formula allows us to convert a logarithm from one base to another. It states that logbx=logcxlogcb where b and c are any two bases.
Let's say we want to evaluate the logarithm log5100. We can use the change of base formula to convert this logarithm to a different base, like base 10:
log5100=log10100log105
Now, we can use a calculator (or our knowledge of logarithms) to evaluate the two base-10 logarithms on the right side of the equation:
log510020.69892.861.

How can we use logarithms to solve exponential equations?

Logarithms can be a really useful tool for solving exponential equations. For example, say we want to solve 2x=9. We can take the logarithm of both sides, and use the properties of logarithms to isolate the variable:
2x=9log102x=log109xlog102=log109x=log109log102x3.167

How are logarithms and radicals used in the real world?

Logarithms and radicals are used in lots of different ways in the real world. They are important in fields like finance, engineering, and science. For example, the Richter scale, which measures the magnitude of earthquakes, uses a logarithmic scale. This means that a magnitude 6 earthquake is actually ten times stronger than a magnitude 5 earthquake!

Want to join the conversation?

  • piceratops ultimate style avatar for user Durgen
    I think there's a typo in "How can we use logarithms to solve exponential equations?". it says "solve 2^x=10" and then it solves 2^x=9. so unless I'm extremely bad at reading numbers, you might want to change that if possible. I don't mean to be rude, but it did confuse me and might proceed to confuse people later on.
    (35 votes)
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  • blobby green style avatar for user kevin123rojas
    on the ection that says "What is the change of base formula for logarithms?" on the next paragraph it says that log_c(x)/log_c(b) for any value of b and c.
    i was wonering if even 0 is included or negative values, because i think with 0 it gets undifined.
    thanks
    (5 votes)
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  • blobby green style avatar for user jianyuwang2006
    for the definition of e, (1+1/n)^n, one thing I don't get it is that since the exponent is n, can we expand the formula into 1^n + 1/n^n, and let n be a huge number like infinity and then this will be 1 + a very small number and the result will be 1 instead of 2.7......? also if we don't expand the formula but still let n be infinity, this will be 1 plus a very small number to a huge power, and I think the result will be close to 1 or 2, why is 2.7...?
    (5 votes)
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    • leaf green style avatar for user Tanner P
      I see where you are coming from. The problem is that (1+1/n)^n does not equal 1^n+(1/n)^n. Remember that you can't just raise everything you're adding in the parentheses to the power. For example, if you want to find (a+b)^2 it is not a^2+b^2, instead it is a^2+2ab+b^2. Hope this clears things up!
      (6 votes)
  • blobby green style avatar for user yahya.sadowski
    This page says that log_9/log_2 = 3.167. My TR-83 calculator says it equals 3.1699... What gives?
    (7 votes)
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    • duskpin tree style avatar for user WolfyNightshade
      Whoever did the rounding for that made a mistake. You were correct to say that it was 3.1699. I have a TI-84 Plus Color Edition graphing calculator. I entered the equation log(9)/log(2) and it answered 3.169925001, confirming that your answer was indeed correct. Rounded to 5 digits, this value is indeed 3.1699. However, rounded to 4 digits, it should have been 3.170. Whoever did the rounding for this put the 7 in the wrong spot.
      (1 vote)
  • blobby green style avatar for user Jeff Liu
    In the last part, how did you get 2^x = 9
    (4 votes)
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    • blobby blue style avatar for user joshua
      Either its a typo in the statement or they accidentally calculated the wrong thing.

      Also, you might want to see other questions before posting yours to avoid making duplicate questions. (It has been raised below)
      (3 votes)
  • blobby green style avatar for user Adyn944
    Why does ln e equal 1, what would a ln e equal and what would ln e^b equal?
    (1 vote)
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  • blobby green style avatar for user Ckc Ckc
    "logarithms are the inverse operation of exponentiation" This seems misworded to me, wouldn't it be more accurate to say logarithms are the inverse function to exponential functions rather than exponentiation as a whole? Can't roots be an inverse too?

    I mean, for stuff like b^x where b is a constant logs work but if its x^b wouldn't you need to use roots instead?
    (1 vote)
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