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## Algebra (all content)

### Course: Algebra (all content) > Unit 11

Lesson 25: The change of base formula for logarithms (Algebra 2 level)- 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 | log, start base, b, end base, left parenthesis, M, N, right parenthesis, equals, log, start base, b, end base, left parenthesis, M, right parenthesis, plus, log, start base, b, end base, left parenthesis, N, right parenthesis | |

Quotient rule | log, start base, b, end base, left parenthesis, start fraction, M, divided by, N, end fraction, right parenthesis, equals, log, start base, b, end base, left parenthesis, M, right parenthesis, minus, log, start base, b, end base, left parenthesis, N, right parenthesis | |

Power rule | log, start base, b, end base, left parenthesis, M, start superscript, p, end superscript, right parenthesis, equals, p, log, start base, b, end base, left parenthesis, M, right parenthesis | |

Change of base rule | log, start base, b, end base, left parenthesis, M, right parenthesis, equals, start fraction, log, start base, a, end base, left parenthesis, M, right parenthesis, divided by, log, start base, a, end base, left parenthesis, b, right parenthesis, end fraction |

*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 log, left parenthesis, 2, x, right parenthesis as log, left parenthesis, 2, right parenthesis, plus, log, left parenthesis, x, right parenthesis. Because the resulting expression is longer, we call this an

**expansion**.In another example, we can use the change of base rule to rewrite start fraction, natural log, left parenthesis, x, right parenthesis, divided by, natural log, left parenthesis, 2, right parenthesis, end fraction as log, start base, 2, end base, left parenthesis, x, right parenthesis. 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 log (which is log base 10) and natural log (which is log base e).

Suppose, for example, we want to evaluate log, start base, 2, end base, left parenthesis, 7, right parenthesis. We can use the change of base rule to rewrite that logarithm as start fraction, natural log, left parenthesis, 7, right parenthesis, divided by, natural log, left parenthesis, 2, right parenthesis, end fraction 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?(18 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(6 votes)

- in
`log_1(1)=x`

, doesn't`x = infinity`

?(8 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(14 votes)

- Why would you need to use ln?(7 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.(7 votes)

- Is ln the same thing as log base 10?(5 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(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)

- How do you do log base 2 x + log base 3 x = 4?(4 votes)
- why e^𝝿i+1=0?

How did Euler proof this equation?(2 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).(3 votes)

- For number one, why can't you '''\frac{\log \left(3\right)+\log \left(a\right)}{\log \left(2\right)}?'''(3 votes)
- What is log_15 (9), if log_15 (5) = a

The answer is: 2-2a

Could someone explain the steps to solve this?(2 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!(2 votes)