# Change of base formula, is this what hes talking about? logs

• mr_coffee
In summary, the conversation discusses the change of base formula in discrete math, where one can convert a logarithm with a certain base to a logarithm with a different base and still evaluate the answer. The use of calculators, tables, and multiple choice questions are also mentioned. The conversation ends with an example of solving a logarithmic equation without a calculator or tables.
mr_coffee
Hello everyone.

This is for a descrete math class, and he said you must know the change of base formula, so if you have everything in log base 3, you can figure out what it is in log base 2 as an example.

But I'm looking up the examples on the interenet and they all seem to convert whatever log they have to log base 10 and evaluate from there. Is this also what he would be talking about? No calculators are allowed to evaluate the logs.

Here is the change of base formula:
http://www.icoachmath.com/SiteMap/images/clip_image012_001.gif

and here is the example I'm looking at:
http://www.icoachmath.com/SiteMap/images/clip_image014_002.gif Okay i see that x = 4, and z = 8, but what is the base y? is the base y the new base you are trying to convert the orginal base into? FOr this example, you are given a base 4 log, and your trying to convert it to a log of base 2 and evaluate the answer I'm assuming right?

how did they end up with 3/2 though?

EDIT: I figured out how they got 3/2, i forgot about the other form of logs, 2^3 = 8, and 2^2 = 4.

Another example:
Use the change of base formula to evaluate
http://www.icoachmath.com/SiteMap/images/clip_image002_084.gif Choices:

A. 4.56

B. 4.86

C. 0.21

D. 5.16

Solution:

Step 1:http://www.icoachmath.com/SiteMap/images/clip_image002_085.gif

Also is it up to you what base you convert it too? I see they used base 10 but there was no base 10 in the orginal problem.
how did they get: 2.43 from log_10(29)/log_10(4) ?I know this is algebra stuff but its been awhile

Thanks~

Last edited by a moderator:
mr_coffee said:
Hello everyone.

This is for a descrete math class, and he said you must know the change of base formula, so if you have everything in log base 3, you can figure out what it is in log base 2 as an example.

But I'm looking up the examples on the interenet and they all seem to convert whatever log they have to log base 10 and evaluate from there. Is this also what he would be talking about? No calculators are allowed to evaluate the logs.

Here is the change of base formula:
http://www.icoachmath.com/SiteMap/images/clip_image012_001.gif

and here is the example I'm looking at:
http://www.icoachmath.com/SiteMap/images/clip_image014_002.gif

Okay i see that x = 4, and z = 8, but what is the base y? is the base y the new base you are trying to convert the orginal base into? FOr this example, you are given a base 4 log, and your trying to convert it to a log of base 2 and evaluate the answer I'm assuming right?

how did they end up with 3/2 though?

EDIT: I figured out how they got 3/2, i forgot about the other form of logs, 2^3 = 8, and 2^2 = 4.

Another example:
Use the change of base formula to evaluate
http://www.icoachmath.com/SiteMap/images/clip_image002_084.gif

Choices:

A. 4.56

B. 4.86

C. 0.21

D. 5.16

Solution:

Step 1:http://www.icoachmath.com/SiteMap/images/clip_image002_085.gif

Also is it up to you what base you convert it too? I see they used base 10 but there was no base 10 in the orginal problem.
how did they get: 2.43 from log_10(29)/log_10(4) ?

I know this is algebra stuff but its been awhile

Thanks~

They choose 10 because most calculators have a log base 10 button, but not a log base 2 button (I'm none have this).

Another choice is base e, which is what I would use personally. All scientific calculators have a button for log base e, and it is usually understood as ln.

That's the real reason.

Sure, you can use base 5, but now what? You still can't solve it.

Note: I saw that you can't use calculators, then I'm assuming you are allowed tables or something to figure it out.

Last edited by a moderator:
Oh, since it is multiple choice, you can just check each answer by brute force. Not that hard.

we won't be allowed to use tables or calculators so maybe i'll switch everytyhing to base 2?

mr_coffee said:
we won't be allowed to use tables or calculators so maybe i'll switch everytyhing to base 2?

No, just solve by brute force. I can hardly call it force though.

You have multiple choices, so you can simply check which is correct one by one.

The solution is probably just to prove that it is the solution.

On the exam it won't be multiple choice though, this was an example i found on the internet to help me study. His exams usually arn't multiple choice but maybe i can hope hah.

Oh!

I see it now.

The one online isn't a good example though because it doesn't have a "nice" solution to it.

The one your teacher gave does have a nice solution. Notice how he appropriately chose 2 as the base?

Here, I made a "nice" solution question for you...

Solve the following without calculator or tables:

$$log_{243}27$$

i got log_3(27)/log_3(243) = 3/5

look good?

BRAVO!

Now, from that exercise, you probably know how I constructed it.

I just picked a number like 6/7 and worked backwards. Once you understand how I did it, you will have no problem solving them in the future.

thanks that did help! :)

## 1. What is the change of base formula for logarithms?

The change of base formula for logarithms states that logb(x) = loga(x) / loga(b), where a, b, and x are positive real numbers with a and b not equal to 1.

## 2. Why is the change of base formula important?

The change of base formula is important because it allows us to evaluate logarithms with bases that are not commonly used, such as base 2 or base 10. It also helps us to convert between different bases, which is useful in many mathematical and scientific calculations.

## 3. How do you use the change of base formula to evaluate a logarithm?

To use the change of base formula, first identify the given logarithm and the desired base. Then, substitute the values into the formula, and solve for the logarithm value using a calculator or by simplifying the logarithmic expression.

## 4. Can the change of base formula be used for natural logarithms?

Yes, the change of base formula can be used for natural logarithms, which have a base of e. In this case, the formula becomes ln(x) = log(x) / log(e) = log(x).

## 5. Are there any limitations to the change of base formula?

The change of base formula is only applicable for positive real numbers and cannot be used for complex numbers. Additionally, it can only be used for logarithms with a base greater than 0 and not equal to 1.

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