Writing logs in exponential form

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Writing logs in exponential form

In convert Exponentials and Logarithms we will mainly discuss how to change the logarithm expression to Exponential expression and conversely from Exponential expression to logarithm expression. To discus about convert Exponentials and Logarithms we need to first recall about logarithm and exponents. The logarithm of any number to a given

base is the index of the power to which the base must be raised in order to equal the given number. Thus, if a = N, x is called the logarithm of N to the base a. For example: 1. Since 3 = 81, the logarithm of 81 to base 3 is 4. 2. Since 10? = 10, 10? = 100, 10? = 1000, ............. The natural number 1, 2, 3, ...... are respectively the logarithms of 10, 100,

1000, ...... to base 10. The logarithm of N to base a is usually written as log N, so that the same meaning is expressed by the two equations ax = N; x = loga N 1. Convert the following exponential form to logarithmic form: (i) 104 = 10000 Solution: 104 = 10000 log10 10000 = 4 (ii) 3-5 = x Solution: 3-5 = x log3 x = -5 (iii) (0.3)3 = 0.027 Solution:

(0.3)3 = 0.027 log0.3 0.027 = 3 2. Convert the following logarithmic form to exponential form: (i) log3 81 = 4 Solution: log3 81 = 4 34 = 81, which is the required exponential form. (ii) log8 32 = 5/3 Solution: log8 32 = 5/3 85/3 = 32 (iii) log10 0.1 = -1 Solution: log10 0.1 = -1 10-1 = 0.1. 3. By converting to exponential form, find the values of

following: (i) log2 16 Solution: Let log2 16 = x 2x = 16 2x = 24 x = 4, Therefore, log2 16 = 4. (ii) log3 (1/3) Solution: Let log3 (1/3) = x 3x = 1/3 3x = 3-1 x = -1, Therefore, log3(1/3) = -1. (iii) log5 0.008 Solution: Let log5 0.008 = x 5x = 0.008 5x = 1/125 5x = 5-3 x = -3, Therefore, log5 0.008 = -3. 4. Solve the following for x: (i)

logx 243 = -5 Solution: logx 243 = -5 x-5 = 243 x-5 = 35 x-5 = (1/3)-5 x = 1/3. (ii) log5 x = 4 Solution: log5 x = 4 x = (5)4 x = (51/2)4 x = 52 x = 25. (iii) logx 8 = 6 Solution: logx 8 = 6 (x)6 = 8 (x1/2)6 = 23 x3 = 23 x = 2. The logarithm function with base a has domain all positive real numbers and is defined by

where M > 0, a > 0, a 1 Logarithmic Form

Exponential Form loga M = x

M = ax Log7 49 = 2

72 = 49 Exponential Form

Logarithmic Form M = ax

loga M = x 24 = 16

log2 16 = 4 10-2 = 0.01

log10 0.01 = -2 81/3 =

2

log8 2 = 1/3 6-1 = 1/6

log6 1/6 = -1 Logarithmic Form

Exponential Form loga M = x

M = ax log2 64 = 6

26 = 64 log4 32 = 5/2

45/2= 32 log1/82 = -1/3

(1/8)-1/3 = 2 log3 81 = x

3x = 81 log5 x = -2

5-2 = x log x = 3

103 = x 1. log5 x = 2 x = 52 = 25 2. log81 x = ? x = 811/2 x= (92)1/2 x = 9 3. log9 x = -1/2 x = 9-1/2 x = (32)-1/2 x = 3-1 x= 1/3 4. log7 x = 0 x= 70 x = 1 1. log3 27 = n 3n = 27 3n = 33 n = 3 2. log10 10,000 = n 10n = 10,000 10n = 104 n = 4 3. log49 1/7 = n 49n = 1/7 (72)n = 7-1 72n = 7-1 2n = -1 n = -1/2 4.

log36 216 = n 36n = 216 (62)n = 63 62n= 63 2n = 3 n = 3/21. logb 27 = 3 b3 = 27 b3 = 33 b = 3 2. logb 4 = 1/2 b1/2 = 4 (b1/2)2 = 42 b = 16 3. logb 8 = -3 b-3 = 8 b-3 = 23 (b-1)3 = 23 b-1 = 2 1/b = 2 b = ? 4. logb 49 = 2 b2 = 49 b2 = 72 b = 7 If f(x) = log3 x, find f(1). Solution: f(1) = log3 1 = 0 (since logarithm

of 1 to any finite non-zero base is zero.) Therefore f(1) = 0 A number that is domain of the function y = log10 x is (a) 1 (b) 0 (c) ? (d) =10 Answer: (b) The graph of y = log4 x lines entirely in quadrants (a) I and II (b) II and III (c) I and III (d) I and IV At what point does the graph of y = log5 x intersect the x-axis? (a) (1, 0) (b) (0, 1) (c) (5, 0) (d)

There is no point of intersection. Answer: (a) Mathematics LogarithmMathematics LogarithmsConvert Exponentials and LogarithmsLogarithm Rules or Log RulesSolved Problems on LogarithmCommon Logarithm and Natural LogarithmAntilogarithm Logarithms 11 and 12 Grade Math From Convert Exponentials and Logarithms to HOME PAGE

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need. Share this page: What's this? The exponent of a number says how many times to use the number in a multiplication. In this example: 23 = 2 ? 2 ? 2 = 8 (2 is used 3 times in a multiplication to get 8) A

Logarithm goes the other way. It asks the question "what exponent produced this?": And answers it like this: In that example: The Exponent takes 2 and 3 and gives 8 (2, used 3 times in a multiplication, makes 8) The Logarithm takes 2 and 8 and gives 3 (2 makes 8 when used 3 times in a multiplication) A Logarithm says how many of one number to

multiply to get another number So a logarithm actually gives you the exponent as its answer: (Also see how Exponents, Roots and Logarithms are related.) Working Together Exponents and Logarithms work well together because they "undo" each other (so long as the base "a" is the same): They are "Inverse Functions" Doing one, then the other, gets

you back to where you started: Doing ax then loga gives you x back again: Doing loga then ax gives you x back again: It is too bad they are written so differently ... it makes things look strange. So it may help to think of ax as "up" and loga(x) as "down": going up, then down, returns you back again:down(up(x)) = x going down, then up, returns you

back again:up(down(x)) = x Anyway, the important thing is that: The Logarithmic Function is "undone" by the Exponential Function. (and vice versa) Like in this example: Start with:log3(x) = 5 We want to "undo" the log3 so we can get "x =" Use the Exponential Function (on both sides): And we know that , so:x = 35 Answer: x = 243 And also: Start

with:y = log4(1/4) Use the Exponential Function on both sides: Simplify:4y = 1/4 Now a simple trick: 1/4 = 4-1 So:4y = 4-1 And so:y = -1 Properties of Logarithms One of the powerful things about Logarithms is that they can turn multiply into add. loga( m ? n ) = logam + logan "the log of multiplication is the sum of the logs" Why is that true? See

Footnote. Using that property and the Laws of Exponents we get these useful properties: loga(m ? n) = logam + logan the log of multiplication is the sum of the logs loga(m/n) = logam - logan the log of division is the difference of the logs loga(1/n) = -logan this just follows on from the previous "division" rule, because loga(1) = 0 loga(mr) =

r ( logam ) the log of m with an exponent r is r times the log of m Remember: the base "a" is always the same! History: Logarithms were very useful before calculators were invented ... for example, instead of multiplying two large numbers, by using logarithms you could turn it into addition (much easier!) And there were books full of Logarithm

tables to help. Let us have some fun using the properties: Start with:loga( (x2+1)4x ) Use loga(mn) = logam + logan :loga( (x2+1)4 ) + loga( x ) Use loga(mr) = r ( logam ) : 4 loga(x2+1) + loga( x ) Also x = x? :4 loga(x2+1) + loga( x? ) Use loga(mr) = r ( logam ) again: 4 loga(x2+1) + ? loga(x) That is as far as we can simplify it ... we can't do

anything with loga(x2+1). Answer: 4 loga(x2+1) + ? loga(x) Note: there is no rule for handling loga(m+n) or loga(m-n) We can also apply the logarithm rules "backwards" to combine logarithms: Start with:loga(5) + loga(x) - loga(2) Use loga(mn) = logam + logan :loga(5x) - loga(2) Use loga(m/n) = logam - logan : loga(5x/2) Answer: loga(5x/2) The

Natural Logarithm and Natural Exponential Functions When the base is e ("Euler's Number" = 2.718281828459...) we get: The Natural Logarithm loge(x) which is more commonly written ln(x) The Natural Exponential Function ex And the same idea that one can "undo" the other is still true: ln(ex) = x e(ln x) = x And here are their graphs: Natural

Logarithm Natural Exponential Function Graph of f(x) = ln(x) Graph of f(x) = ex Passes through (1,0) and (e,1) Passes through (0,1) and (1,e) They are the same curve with x-axis and y-axis flipped. Which is another thing to show you they are inverse functions. On a calculator the Natural Logarithm is the "ln" button. Always try to use Natural

Logarithms and the Natural Exponential Function whenever possible. The Common Logarithm When the base is 10 you get: The Common Logarithm log10(x), which is sometimes written as log(x) Engineers love to use it, but it is not used much in mathematics. On a calculator the Common Logarithm is the "log" button. It is handy because it tells you

how "big" the number is in decimal (how many times you need to use 10 in a multiplication). Well, 10 ? 10 = 100, so when 10 is used 2 times in a multiplication you get 100: log10 100 = 2 Likewise log10 1,000 = 3, log10 10,000 = 4, and so on. OK, best to use my calculator's "log" button: log10 369 = 2.567... Changing the Base What if we want to

change the base of a logarithm? Easy! Just use this formula: "x goes up, a goes down" Or another way to think of it is that logb a is like a "conversion factor" (same formula as above): So now we can convert from any base to any other base. Another useful property is: See how "x" and "a" swap positions? 1 / log8 2 = log2 8 And 2 ? 2 ? 2 = 8, so when 2

is used 3 times in a multiplication you get 8: 1 / log8 2 = log2 8 = 3 But we use the Natural Logarithm more often, so this is worth remembering: My calculator doesn't have a "log4" button ... ... but it does have an "ln" button, so we can use that: log4 22 = ln 22 / ln 4 = 3.09.../1.39... = 2.23 (to 2 decimal places) What does this answer mean? It means

that 4 with an exponent of 2.23 equals 22. So we can check that answer: Check: 42.23 = 22.01 (close enough!) Here is another example: log5 125 = ln 125 / ln 5 = 4.83.../1.61... =3 (exactly) I happen to know that 5 ? 5 ? 5 = 125, (5 is used 3 times to get 125), so I expected an answer of 3, and it worked! Real World Usage Here are some uses for

Logarithms in the real world: Earthquakes The magnitude of an earthquake is a Logarithmic scale. The famous "Richter Scale" uses this formula: M = log10 A + B Where A is the amplitude (in mm) measured by the Seismograph and B is a distance correction factor Nowadays there are more complicated formulas, but they still use a logarithmic scale.

Sound Loudness is measured in Decibels (dB for short): Loudness in dB = 10 log10 (p ? 1012) where p is the sound pressure. Acidic or Alkaline Acidity (or Alkalinity) is measured in pH: pH = -log10 [H+] where H+ is the molar concentration of dissolved hydrogen ions. Note: in chemistry [ ] means molar concentration (moles per liter). More

Examples Start with:2 log8 x = log8 16 Bring the "2" into the log:log8 x2 = log8 16 Remove the logs (they are same base): x2 = 16 Solve:x = -4 or +4 But ... but ... but ... you cannot have a log of a negative number! So the -4 case is not defined. Answer: 4 Check: use your calculator to see if this is the right answer ... also try the "-4" case. Start

with:e-w = e2w+6 Apply ln to both sides:ln(e-w) = ln(e2w+6) And ln(ew)=w: -w = 2w+6 Simplify:-3w = 6 Solve:w = 6/-3 = -2 Answer: w = -2 Check: e-(-2)= e2 and e2(-2)+6=e2 To see why, we will use and : First, make m and n into "exponents of logarithms": Then use one of the Laws of Exponents Finally undo the exponents. It is one of

those clever things we do in mathematics which can be described as "we can't do it here, so let's go over there, then do it, then come back" Copyright ? 2017 LOGARITHMIC FUNCTIONS A logarithmic function is the inverse of an exponential function. An inverse function is obtained by interchanging the x and y in a function. A

logarithmic function can be written in the form x=ay In the previous section we graphed equations of this form. Rewriting in log form: When we write a log (or logarithmic) function as x=ay we say it is in exponential form. The alternative is logarithmic form written y = loga x which is read as "log of x base a" or "log base a of x." Definition of a log:

loga x is defined as the exponent needed to raise a to in order to get x. log3 81 is equal to 4 since 3 must be raised to the 4th power to get 81 log3 3m is equal to m since 3 must be raised to the mth power in order to get 3m logx x is equal to 1 since x must be raised to the 1st power in order to get x logx 1 is equal to 0 since x must be raised to the

zero power in order to get 1 Converting between exponential and logarithmic form: x=ay and y = loga x are both stating the same relationship between x and y. They both indicate that if you raise a to the y power you will get x. We need to practice converting from one form to the other. Log form to exponential form: The base for both forms is a. To

do this conversion, start with the base. The exponent for the base will be on the other side of the equal. Next put in an equal sign and write the expression on the other side of the equal sign. Exponential form to log form: Start with the word log and the base as the subscript. Put the exponent on the other side of an equal sign. (The log is equal to the

exponent.) Now complete the log with the expression. log form exponential form log2 8 = 3 23 = 8 logp w = c pc = w log101000 = 3 103 = 1000 log10 0.1 = -1 10-1 = 0 .1 Solving Simple Logarithmic Equations: On equations of the form y = loga x where a or x are the unknown, it is possible to solve for x by changing to exponential form. Solve for

x: log5 x = 3 Changing to exponential form gives the solution x = 53 or x = 125 Solve for x: logx 25 = 2 Changing to exponential form gives x2=25 Since the base must be positive, the only solution is x=5. Finding Logs on a calculator:Logs base 10 are called common logs and when a base is not written, it is assumed to be base 10. Log 100

means log10100 or 2. On a calculator there is a log button which will give logs base 10 of any number.

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