- Evaluating logarithms: change of base rule
- Logarithm change of base rule intro
- Evaluate logarithms: change of base rule
- Using the logarithm change of base rule
- Use the logarithm change of base rule
- Proof of the logarithm change of base rule
- Logarithm properties review
Review the logarithm properties and how to apply them to solve problems.
What are the logarithm properties?
|Change of base rule|
Want to learn more about logarithm properties? Check out this video.
Rewriting expressions with the properties
We can use the logarithm properties to rewrite logarithmic expressions in equivalent forms.
For example, we can use the product rule to rewrite
as . Because the resulting expression is longer, we call this an expansion.
In another example, we can use the change of base rule to rewrite
as . Because the resulting expression is shorter, we call this a compression.
Evaluating logarithms with calculator
Calculators usually only calculate
(which is log base ) and (which is log base ).
Suppose, for example, we want to evaluate
. We can use the change of base rule to rewrite that logarithm as and then evaluate in the calculator:
Round your answer to the nearest thousandth.
Round your answer to the nearest thousandth.
Want to join the conversation?
- How did a mathematician find e? What's its origin?(19 votes)
- It is often called Euler's number after Leonhard Euler (pronounced "Oiler")
e is an irrational number (it cannot be written as a simple fraction).
e is the base of the Natural Logarithms (invented by John Napier).
e is found in many interesting areas, so is worth learning about. you can check this link to find out:
https://www.khanacademy.org/math/in-in-grade-11-ncert/in-in-exponential-and-logarithmic-functions/copy-of-math3-e-and-natural-log/v/e-through-compound-interest and https://www.khanacademy.org/math/in-in-grade-11-ncert/in-in-exponential-and-logarithmic-functions/copy-of-math3-e-and-natural-log/v/e-as-limit(7 votes)
x = infinity?(8 votes)
- See the "Restrictions" section at this link (about 1/2 down page). The base is restricted from being 1.
- Why would you need to use ln?(8 votes)
- The natural log function, ln, is the log with a base of Euler's number, e.
Here is an example of when it can be used:
e^x = 2
--> To solve for x, we would take the ln of both sides. This is because x is the exponent of e, and the e and natural log will cancel out when put together.
ln(e^x) = ln(2)
x = ln(2)
This is the most common way I've seen the natural log used, but there are no doubt other ways to use it.(8 votes)
- Is ln the same thing as log base 10?(6 votes)
- The "log" key on a calculator is log base 10. "ln" is natural logarithm, and there is a video for that here: https://www.khanacademy.org/math/algebra2/exponential-and-logarithmic-functions/e-and-the-natural-logarithm/v/natural-logarithm-with-a-calculator(7 votes)
- why e^𝝿i+1=0?
How did Euler proof this equation?(6 votes)
- It is a specific case of his formula, e^(i*x) = cos(x) + i*sin(x). The proof of this formula requires Calculus level math though (e.g. power series or differentiation).(4 votes)
- What is log_15 (9), if log_15 (5) = a
The answer is: 2-2a
Could someone explain the steps to solve this?(5 votes)
- I can explain how to check it, at least, though I'm not sure that this is how you would originally come to this answer.
log_15(5)=a, and we want to see whether log_15(9)=2-2a. We can begin by finding a.
log(5)/log(15)=a Use the change of base rule.
Next we can find 2-2a:
2-2*approx.0.5943 Try to use the entire answer you got for a, instead of a rounded one, if you can.
2-approx. 1.1886=approx. 0.8114
Now we can find log_15(9), and see if it equals 0.8114.
log_15(9)=log(9)/log(15) Change of base rule.
This shows that it is indeed the case that if log_15(5)=a, then log_15(9)=2-2a, but it does not seem like it is the way that you would come to figure it in the first place.
Perhaps if you figured that a=approx. 0.5943, and that log_15(9)=approx. 0.8114, you might just happen to notice that 0.8114+2(0.5943)=approx. 2. I have not figured anything better than this for this question. Maybe someone else will. For now, I hope you have a good day. Keep going!(4 votes)
- Why is the base 10 logarithmic scale the standard for calculators?(1 vote)
- Probably because the rest of our number system is built around powers of 10 --- tens, hundreds, thousands, etc. and tenths, hundredths, thousandths, etc.(7 votes)
- e is the base of the Natural Logarithms (invented by John Napier), how did he did it and did he calculated the logarithms by hand?(3 votes)