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### Course: Algebra 2 > Unit 8

Lesson 4: The change of base formula for logarithms- 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

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# Logarithm properties review

Review the logarithm properties and how to apply them to solve problems.

## What are the logarithm properties?

Product rule | ||

Quotient rule | ||

Power rule | ||

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 $\mathrm{log}(2x)$ as $\mathrm{log}(2)+\mathrm{log}(x)$ . Because the resulting expression is longer, we call this an

**expansion**.In another example, we can use the change of base rule to rewrite $\frac{\mathrm{ln}(x)}{\mathrm{ln}(2)}$ as ${\mathrm{log}}_{2}(x)$ . Because the resulting expression is shorter, we call this a

**compression**.*Want to try more problems like this? Check out this exercise.*

## Evaluating logarithms with calculator

Calculators usually only calculate $\mathrm{log}$ (which is log base $10$ ) and $\mathrm{ln}$ (which is log base $e$ ).

Suppose, for example, we want to evaluate ${\mathrm{log}}_{2}(7)$ . We can use the change of base rule to rewrite that logarithm as $\frac{\mathrm{ln}(7)}{\mathrm{ln}(2)}$ and then evaluate in the calculator:

*Want to try more problems like this? Check out this exercise.*

## Want to join the conversation?

- How did a mathematician find e? What's its origin?(23 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(12 votes)

- in
`log_1(1)=x`

, doesn't`x = infinity`

?(9 votes)- See the "Restrictions" section at this link (about 1/2 down page). The base is restricted from being 1.

https://www.khanacademy.org/math/algebra2/exponential-and-logarithmic-functions/introduction-to-logarithms/a/intro-to-logarithms(18 votes)

- Why would you need to use ln?(9 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.(9 votes)

- why e^𝝿i+1=0?

How did Euler proof this equation?(8 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).(6 votes)

- Is ln the same thing as log base 10?(7 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)

- What is log_15 (9), if log_15 (5) = a

The answer is: 2-2a

Could someone explain the steps to solve this?(7 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_15(5)=a

log(5)/log(15)=a Use the change of base rule.

a=approx. 0.5943

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.

log(9)/log(15)=approx. 0.8114

log_15(9)=2-2a True.

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!(5 votes)

- How do you do log base 2 x + log base 3 x = 4?(6 votes)
- Can anyone explain to me how to solve e^ln^2 x +x^lnx =2e^4(5 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)