Question

solve log11 (y+8) + log11 4 = log11 60

Answer 1

@mathmale

Answer 2

\[\log_{11} (y+8) + \log_{11} (4) = \log_{11} (60)\]

First thing to remember is that \[\log_a b + \log_a c = \log_a (b*c)\]

Answer 3

If you apply that, what does the left hand side become?

Answer 4

a is 11 b is y + 8 and c is 4 correct ?

Answer 5

sure, if you want. The general notion is that the sum of the logs of two quantities is the log of the product of the two quantities.

Answer 6

\[\log_{11} (y+8)(4)\]

Answer 7

To be clear, \[\log_{11} (4(y+8))\]

Now the right hand side is unchanged, so we have \[\log_{11} (4(y+8)) = \log_{11} (60)\]What does that imply about the relationship between \(4(y+8)\) and \(60\)?

Answer 8

im not sure?

Answer 9

Doesn't it mean they have to be equal?

Doesn't \[\log x = \log y\] mean that \(x = y\)?

Answer 10

oh. yea

Answer 11

so now we find x and y?

Answer 12

\[4(y+8) = 60\]

Answer 13

solve for y

Answer 14

yes, please.

Answer 15

ok x = 8

Answer 16

try again please :-)

Answer 17

6 ***

Answer 18

we want to find \(y\) such that \(4(y+8) = 60\) No other letters apply in this problem, the use of \(x\) was just as a placeholder in the example...it could have as easily been \[\log c = \log d\]

Answer 19

ok so does x = 6?

Answer 20

there is no \(x\) in this problem!

Answer 21

so what is y ?

Answer 22

and whatever you think your next answer is, try it in \(4(y+8) = 60\) before you post it.

Answer 23

Solve the equation
\[4(y+8) = 60\]for \(y\). What do you get?

Answer 24

y = 7

Answer 25

Yes. That is the answer.

Answer 26

thanks!!

Answer 27

Any questions about how we got there?

Answer 28

just a little confused about how we set that equation up

Answer 29

Which part, converting the addition of logs into the log of a product, or solving from there on?

Answer 30

converting the addition

Answer 31

It is 7

Answer 32

4(y + 8) = 60
4y + 32 = 60
4y = 28
y = 7

Answer 33

so, log11 (y + 8) + log11 4 = log11 60 it can rewriten be log11 (y + 8)4 = log11 60 now, cancel out the log 11 to both sides, giving us (y+8)4 = 60 solve for y so it will equal 7

Answer 34

Well, let's back up a bit. Do you understand that a logarithm is just an exponent?

Answer 35

i think thats correct

Answer 36

yes

Answer 37

Okay, so let's make it easy for calculating and use 2 as the base of our logarithms for this example. Let's say we have \[\log_2 4 + \log_2 8\]The logarithm base 2 of 4 is the number that we raise the base to get 4. 2^2 = 4. The logarithm base 2 of 8 is the number that we raise the base to get 8. 2^3 = 8. Okay so far?

Answer 38

yes..

Answer 39

Okay, let's add those two together: \(2+3 = 5\)

Now, say we have a number \(x\) whose logarithm base 2 happens to be \(5\). Can you find the value of \(x\)?
\[\log_2 x = 5\]\[x=\]

Answer 40

Remember, if \[\log_2 x = 5\]that implies that \[2^5 = x\]

Answer 41

yes i understand that

Answer 42

Okay, so \(x =\)

Answer 43

32

Answer 44

Right.

What does \(4*8=\)

Answer 45

32

Answer 46

Right. So \[\log_2 4 + \log_2 8 = 2+ 3 = 5\]and\[\log_2 32 = \log_2 (4*8) = \log_2 4 + \log_2 8 = 5\]This is no accident!

Answer 47

\[b^m*b^n = b^{m+n}\]

That's what we are doing when we add logarithms of a common base.

Answer 48

ahh

Answer 49

In our example, \[2^m*2^n = 2^2*2^3 = 2^{2+3} = 2^5\]

Answer 50

Hopefully, it is also clear that subtracting logarithms is like dividing.

Answer 51

i see now :D thanks so much!

Answer 52

This is how old-school slide rules work — they have scales with logarithmic graduations, and the mechanics of the slide rule are just adding and subtracting lengths.

Answer 53

In the old days, people used to publish books of tables of logarithms to a handful of decimal places. If you wanted to compute \(3.972^{4.026}\), you would look up the logarithm of \(3.972 \approx 0.599009\), multiply it by the exponent to get \(2.41161\), and then look that up in the table that goes from logarithms back to numbers to get \(257.995\).

Answer 54

i see

Answer 55

This also leads to a bad math joke:

Noah's boat finally comes to rest as the flood waters recede, and he lowers the gangway and send the animals out calling to them, "Go forth and multiply".

Most of the animals leave, but two snakes are left behind. Noah looks at them, and commands "Go forth and multiply!"

The snakes look at him but do not move. He tries again, "Go forth and multiply!" The snakes do not move.

Noah gets angry and in his most commanding voice shouts, "Go forth and multiply!"

The snakes look up at him and say, "We can't, we're adders".

Noah thinks for a while, then grabs his saw and hammer and runs off into the forest, where he cuts down a tree. He saws and hammers and builds a small table. He carefully picks up the snakes and puts them on the table.

"Go forth and multiply!" he commands.

The snakes look at each other, and then at Noah. "We can't, we're adders".

"Yes", Noah replies, "but, even adders can multiply on a log table".

Answer 56

LOL i don't get it but i will eventually

Answer 57

Well, the key is that if you have log tables, you can multiply by adding. Add the logs of the terms in the product, take the anti-log to find the answer.