## Appendix N: Derivation of the Logarithm Change of Base Formula

We set out to prove the logarithm change of base formula: logb x = loga x loga b. To do so we let y = logb x and apply these as exponents on the base.

## MATHEMATICS 0110A CHANGE OF BASE Suppose that we have

Let y = logb a. Then we know that this means that by = a. We can take logarithms to base c

Change of Base

## Lesson 5-2 - Using Properties and the Change of Base Formula

Common logarithin and natural logarithm functions are typically built into calculator systems. However it is possible to use a calculator to evaluate.

## 1 Solutions to Homework Exercises : Change of Base Handout

log 8 log 3. (d) For this we want to simplify before we use the formula. after we change to base 2

Sol ChangeBase

## Logarithms - changing the base

This leaflet gives this formula and shows how to use it. A formula for change of base. Suppose we want to calculate a logarithm to base 2. The formula states.

mc logs

## Change of Base Formula.pdf

The Change of Base Formula. Use a calculator to approximate each to the nearest thousandth. 1) log3. 3.3. 2) log2. 30. 3) log4. 5. 4) log2. 2.1. 5) log 3.55.

Change of Base Formula

## 6.11 Notes – Change of base and log equations

Objectives: 1) Use common logs to solve equations. 2) Apply the change of base formula. 1).

day notes . notes change of base keyed

## Untitled

Learning Targets: • Apply the properties of logarithms in any base. ⚫ Compare and expand logarithmic expressions. Use the Change of Base Formula. SUGGESTED

## Change-of-Base Formula. For any logarithmic bases a and b and

Problem #1. Use your calculator to find the following logarithms. Show your work with Change-of-Base Formula. a) b). 2 log 10. 1. 3 log 9 c). 7 log 11.

Lecture

## Properties of exponents Properties of Logarithms The natural

We also use log base 10 very often so we abbreviate that as log10(x) = log(x). Your calculator follows the same convention. Change of Base Formula.

Log Exponential Prop

### Question 1

(a) We simply use the formula: log#### 211 =log311

log32 (b) log#### 57 =log27

log25 (c) This time, we use the laws of logarithms to do some simplifying after using the formula: log 1#### 38 =log8log13=log8log1-log3=log8-log3=-log8log3

(d) For this, we want to simplify before we use the formula. Werecognize that we can switch from base 6 to

base 31.1 log631= log316 =log56log531 (e) This is similar to the previous one. Notice that since log axis defined thenxmust be positive. And since log axis in the denominator, thenxcannot be 1, because logb1 = 0 for any baseb. Thereforex >0 and x?= 1, soxcan be the base of logarithms. We get: 1 logax= logxa=log2alog2x (f) Again, we simplify before changing to base 2. This time, we recognize the formlogba logbc= logca. And after we change to base 2, we use laws of logarithms to simplify. We get: log36 log16= log1636 =log236log216=log2(4×9)log224=(log24) + (log29)4 (log222) + (log232)#### 4=2 + 2log234=1 + log232=12+log232

1#### 2+?12?

log#### 23 =12+ log231/2=12+ log2⎷3

### Question 2

(a) We use the change of base formula, changing to basee, i.e., to ln. log#### 542 =ln42

ln5(b) This time, we switch the base before we change to natural logarithms. Of course, the base of the given

logarithm is 10.1 log23= log2310 =ln10ln23 2 (c) Now, we have a 5 multiplier to contend with: 5 log210= 5?1log210? = 5log2 = 5?ln2ln10? =5ln2ln10=ln25ln10=ln32ln10 (d) And this time the multiplier is disguised: log#### 3⎷

e= log3e1/2=?12? log#### 3e=?12??

lneln3? =?12?? 1ln3? =12ln3=1ln32=1ln9### Question 3

(a) This is easier if we switch from base 25 to base 5: log#### 255 =1

log525=1log552=12 (b) Again, it"s easiest to switch: log#### 642 =1

log264=1log282=12log28=12log223=12(3)=16(c) We recognize that 81 is a power of 3 (i.e., is a power of 9, which is a power of 3) and so is 27, so we

change to base 3: log#### 8127 =log327

log381=log333log392=3log3(32)2=3log332(2)=3log334=34 (d) This time, the base which both 125 and 25 are powers of is 5: log#### 12525 =log525

log5125=log552log5(5)(25)=2log553=23 (e) Here, the base is 1#### 4= 2-2and the number whose logarithm we"re taking is 8 = 23. Therefore base 2 will

be useful. log#### 1/48 =log28

log214=log223log22-2=3-2=-32### Question 4

(a) We start by switching both to base 5, then simplify using laws of logarithms: 1 log25+1log35= log52 + log53 = log5(2×3) = log56 We can return this to a form similar to what we started with by switching to base 6: log56 =1 log65. (b) We use the same approach as in the previous question: 1 1 Solutions to Homework Exercises : Change of Base Handout### Question 1

(a) We simply use the formula: log#### 211 =log311

log32 (b) log#### 57 =log27

log25 (c) This time, we use the laws of logarithms to do some simplifying after using the formula: log 1#### 38 =log8log13=log8log1-log3=log8-log3=-log8log3

(d) For this, we want to simplify before we use the formula. Werecognize that we can switch from base 6 to

base 31.1 log631= log316 =log56log531 (e) This is similar to the previous one. Notice that since log axis defined thenxmust be positive. And since log axis in the denominator, thenxcannot be 1, because logb1 = 0 for any baseb. Thereforex >0 and x?= 1, soxcan be the base of logarithms. We get: 1 logax= logxa=log2alog2x (f) Again, we simplify before changing to base 2. This time, we recognize the formlogba logbc= logca. And after we change to base 2, we use laws of logarithms to simplify. We get: log36 log16= log1636 =log236log216=log2(4×9)log224=(log24) + (log29)4 (log222) + (log232)#### 4=2 + 2log234=1 + log232=12+log232

1#### 2+?12?

log#### 23 =12+ log231/2=12+ log2⎷3

### Question 2

(a) We use the change of base formula, changing to basee, i.e., to ln. log#### 542 =ln42

ln5(b) This time, we switch the base before we change to natural logarithms. Of course, the base of the given

logarithm is 10.1 log23= log2310 =ln10ln23 2 (c) Now, we have a 5 multiplier to contend with: 5 log210= 5?1log210? = 5log2 = 5?ln2ln10? =5ln2ln10=ln25ln10=ln32ln10 (d) And this time the multiplier is disguised: log#### 3⎷

e= log3e1/2=?12? log#### 3e=?12??

lneln3? =?12?? 1ln3? =12ln3=1ln32=1ln9### Question 3

(a) This is easier if we switch from base 25 to base 5: log#### 255 =1

log525=1log552=12 (b) Again, it"s easiest to switch: log#### 642 =1

log264=1log282=12log28=12log223=12(3)=16(c) We recognize that 81 is a power of 3 (i.e., is a power of 9, which is a power of 3) and so is 27, so we

change to base 3: log#### 8127 =log327

log381=log333log392=3log3(32)2=3log332(2)=3log334=34 (d) This time, the base which both 125 and 25 are powers of is 5: log#### 12525 =log525

log5125=log552log5(5)(25)=2log553=23 (e) Here, the base is 1#### 4= 2-2and the number whose logarithm we"re taking is 8 = 23. Therefore base 2 will

be useful. log#### 1/48 =log28

log214=log223log22-2=3-2=-32### Question 4

(a) We start by switching both to base 5, then simplify using laws of logarithms: 1 log25+1log35= log52 + log53 = log5(2×3) = log56 We can return this to a form similar to what we started with by switching to base 6: log56 =1 log65. (b) We use the same approach as in the previous question: 1- logarithm change of base formula proof
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