### Logarithms; What Fun

Can you sense my sarcasm? Oh, logarithms aren't so bad, really. I mean no sarcasm. We just need practice right? Well, what's a better way to start the week, than with a one period math class? We started class with eight questions given to us on the board; all to solve for x.

a) 2

^{x}= 3

b) 4

^{3x}= 2

c) u

^{x}= v

d) a = bc

^{x}

e) log

_{5}x = -2

f) log

_{3}x = π

g) y = log

_{4}3x

h) v = log

_{b}cx

ANSWERS:

a) x = log

_{2}3

**OR**log3/log2

How we get log3/log2 :

2

^{x}= 3

log2

^{x}= log3

xlog2 = log3

x = log3/log2 [divide each side by log2]

b)The way I solved this problem was to change 4 as a power of 2.

2

^{2(3x)}= 2

2(3x) = 1

6x = 1

x = 1/6

Mr. K, then showed us other ways to solve b.

log

_{4}2 = 3x

(1/3)log

_{4}2 = 3x(1/3)

(1/3)log

_{4}2 = x

log

_{4}2

^{(1/3)}= x

log

_{4}

^{3}√2 = x

**OR**

4

^{3x}= 2

log4

^{3x}= log2

3xlog4 = log2

(1/3)3x = log2/log4(1/3)

x = log2/3log4

x = log2/log4

^{3}

c) x = log

_{u}v

**OR**logv/logu [it's exactly like question a), but with letters instead of numbers]

d) a/b = c

^{x}

log

_{c}(a/b) = x

log

_{c}a - log

_{c}b = x

e) log5x = -2

x = 5

^{-}

^{2}

x = 1/25

f) x = 3

^{π}

g) y = log

_{4}3x

log

_{4}3x = y [I just switch sides because it seems less confusing for me with the x on the left side]

3x = 4

^{y}

x = 4

^{y}/3

h) logb

^{cx}= v

cx = b

^{v}

x = b

^{v}/c

So then we were given 3 questions to simplify:

a) 7

^{log74}

b) 2

^{log25}

c) 4

^{log2[2(log25)]}

So first, we always have to remember

**a logarithm is an exponent**.

**Let's use a different example. If given log**

_{2}32, we'd know that the exponent is 5. 32 would be the result of 2(our base number) to the power of an exponent, and in this case it would be 5.

2

^{5}= 32

Then we can say that log

_{2}32 is just another way to say 5. So if we were given 2

^{log232}, it would be like saying 2

^{5}, which is 32.

2^{log232} = 2^{5}

So if given different numbers, like 3^{log35}, it would be stating that 3 to the power of a certain number, will result in 5.

3^{x} = 5

So, 3^{x} is *a certain number itself*, which is equivalent to log_{3}5 OR x, and will result in getting 5 in this given question.

If 3^{x} = 5 then log_{3}5 = x **THEREFORE** 3^{log35} = 5 is the same as 3^{x} = 5

I think that's how it goes. I hope I explained that right, but i think I may have just confused you guys even more. If that's still a bit vague, you can always just **distiguish the pattern**. If the base number is equilvalent to the base number attached to log, then the bigger number [sorry, I don't know the proper terminology], will be your answer.

a^{logab} = b

EX. 6^{log68} = 8

So, the answers would be:

a) 7^{log74} = 4

b) 2^{log25 }= 5

c) 4^{log2[2(log25)] }= 4^{log25} [2^{log25} is just 5, using the pattern we

identified]

= (2^{2})^{(log25)} [we change 4 as a power of 2]

=(2^{2log25})^{2}

=2^{log225}

=25

After that, we were given:

CHANGE OF BASE LAW

log_{a}m = log_{c}m/log_{c}a

EX. log_{3}7 = log7/log3

=ln7/ln3

Mr. K then gave us two questions to work on, but we only had time to answer one.

1. log (x+6) + log(x-6) = 2

log (x+6)(x-6) = 2

log(x^{2}-36) = 2

10^{2} = x^{2}-36 [I'm not 100% sure if that's how they relate or not]

100 = x^{2}-36

0 = x^{2}-136

0 = (x-√136)(x+√136)

x = √136 [accept] AND x = -√136 [reject]

2. [We didn't get to finish this question]

1/2log_{a}(x+2) + 1/2log_{a}(x-2) = 2/3log_{a}27

Our homework is seven questions that's posted on the blog. Yay, I'm done my scribe! Marc, you're the next scribe. Have fun with covering two classes. =)

^{}

## 3 Comments:

Great scribe post Jessica! You used colour in a meaningful way to explain a pattern that is difficult to recognize. You also gave very detailed notes and step-by-step solutions to the problems we did in class today.

For all this your scribe post has been inducted into The Scribe Post Hall Of Fame.

Congratulations!

BTW, the green text:

x = 3^x = log(base 3)5

Should be written:

If 3^x = 5 then log(base 3)5 = x

It's an easy fix if you want to go back and edit it. Especially for a Hall of Famer like you. ;-)

Great post. I like your use of colour to seperate the ideas and concepts. Since this unit does not create concepts that could be illustrated in paint could you find websites out there on the web that reinforce the topic you are scribing about. A practice quiz or more info about logs (not the tree variety:-)

Sometimes it is hard to find a way to spice up a scribe. Your use of colour was great...is there more that you could do???

Mr. Harbeck

Sargent Park School

Hi Jessica,

Congratulations on your Hall of Fame post!!

Thanks for taking time to use such detail in your explanations!

Best,

Lani

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