BML Study Guide: 11th Grade, Logarithmic/Exponential

Equations

Joshua Kim

Meet 2, December 3, 2014

>Took minimal effort>Enjoy and blame dohyun cheon for any mistakes

Let us begin with a definition: For any real a, b, and c, if ab = c, then loga c = b.This is the definition of a logarithm. The latter expression is read as ”the log base a of c equals b”.

Example 1: Evaluate log27 243.Solution 1: Turning this expression into exponential form results in:

27x = 243

A careful reader will notice that 27 is 33 and 243 is 35; therefore, this expression simplifies out into:

33x = 35

The exponents have to equal; therefore, x =5

3

Here are the exponential properties, most of them from alg 2 or precalc:

ab ∗ ac = ab+c

ab

ac= ab−c

(ab)c = ab∗c

b√ac = a

cb

Using those properties, you can prove most, if not all, of the logarithmic properties presented below.

(loga b)(logb c) = loga c

logb c

logb a= loga c

loga b + loga c = loga bc

loga b− loga c = loga

b

cloga b

c = c loga b

loga b =logc b

logc a

loga b =1

logb a

1

The last one on the list is not common and most likely not necessary for competition, but its better toknow than not know. It may help, I don’t know.

Example 2: Prove that xlogx a = a.Solution 2: This is a straightforward application of the definition of a logarithm. Plug in the values andsee if it works yourself.

Also an interesting fact: logx xa = a. You can use this fact to prove that the logarithmic function and the

exponential function are inverses of each other.

There are two main types of logarithms: a common logarithm is a logarithm with base 10 (signifiedsimply by log b), and a natural logarithm is a logarithm with base e (signified by ln b). e, also known asEuler’s number, is approximately 2.71828182846... Derivations of this number should be easily found in anycalculus book.

The change of base formula is basically equivalent to one of the logarithm properties. It allows you totake any logarithm and turn that into an expression you can plug into your calculator (most four-functionand scientific calculators have buttons for common and natural logs).

loga b =log b

log a=

ln b

ln a

Practice time! (There will be more exercises to practice from at the end of this worksheet)

Example 3: Solve the following equation (for all real x) =⇒ log4 x + log4(x + 2) =3

2Solution 3: It’s all log properties and changing the expression to exponential form.

log4 x + log4(x + 2) =3

2=⇒ log4 x(x + 2) =

3

2=⇒ x(x + 2) = 8

=⇒ x = −4, 2

However, this is not our final answer. log4(−4) doesn’t make sense, because there is no value which 4 can

be raised, to result in negative 4. Our only answer is 2 .

This example problem raises up a good point: loga b = c has no solution if a, b, and c are real and b isnegative. LOOK OUT FOR THESE EXTRANEOUS SOLUTIONS! (BML can shaft you quite easily forthese)

Example 4: Solve for x =⇒ 9(x2−5x−5) = 27(x

2+5)

Solution 4: This doesn’t even have to be a logarithm problem. Keep your eyes peeled for the crucial step.

9(x2−5x−5) = 27(x

2+5) =⇒ 32(x2−5x−5) = 33(x

2+5)

=⇒ 3(2x2−10x−10) = 3(3x

2+15)

=⇒ 2x2 − 10x− 10 = 3x2 + 15

=⇒ 0 = x2 + 10x + 25

=⇒ x = −5

No extraneous solutions here, life is good :)

2

Example 5: During a Science Bowl meeting, Daw tells Cowbo to find a number a such that three to thepower of that number will equal to 41. Ricardo, being the master ruseman he is, calculates the numericalvalue of three to the power of three taken from twice that number and begins to yell out, ”The answer is...” What number did Cowbo hear Ricardo say? Express your answer as a common fraction.

Solution 5: I will do this problem two ways, especially since this type of problem comes up often in manycompetitions (of course, not in this form lol). First, exponents.

3a = 41

32a−3 =32a

33=

(3a)2

27

=(41)2

27

=1681

27

Second, logarithms.

3a = 41 =⇒ log3 41 = a

2a− 3 =⇒ 2 log3 41− 3

=⇒ log3 412 − log3 33

=⇒ log3

412

27

32a−3 =⇒ 3log412

27

=⇒ 412

27

=⇒ 1681

27

And that should be it. Have fun with the exercises! Starred ones are notably more difficult than the others.

Exercises

1. Solve for x:3x

9(x+1)= 81(x+2)

2. Simplify to lowest terms:1

log3 2+

1

log9 2

3. If log 2 = 0.301 and log 3 = 0.477, find, to three decimal places, log 288.

4. Simplify to lowest terms:ln 64

2 ln 2

5. log1 1 + log2 1 + log3 1 + · · · log8192 1 = ?

6. In how many distinct points do the lines y = 3 log x and y = log 3x intersect? (Based off of a problemon AHSME 1962)

7. Find all of the solutions of x2 log x =x2

25. (Based off of a problem on AHSME 1978)

3

8. ** If 60a = 3 and 60b = 5, then find 12[(1−a−b)/2(1−b)]. (AHSME 1983)

9. If loga

b+ log

b

a= log(a + b), find a2 + b2 + 2ab.

10. ** Find the sum of all values of x that satisfy the equation log2 x +3

log2 x + 3log2 x+ 3

log2 x+···

= 4

11. Find the value ofx

yif 2 log5(x− 3y) = log5(2x) +

1

log2y 5.

12. The function y = axn passes through the points (2, 1) and (32, 4). Finda

n.

13. ** If logy 8 + log9 x = 2 and2

log3 x+

log2 y

3=

8

3, find the product of the possible values of x.

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