Question

logb(square root x/z^4(y^3) use properties of logarithms to expand the expression?

Answer 1

\[\log_b{\sqrt{\frac{x}{z^4y^3}}}\]
^Like that?

Answer 2

apprentices after b, square root only on x, y^3 separated from z^4 to the right

Answer 3

jabberrwock there?

Answer 4

Like this?
\[\log_b(\frac{\sqrt{x}}{z^4}y^3)\]

Answer 5

yes definitely :)

Answer 6

The first thing to do is recognize that
\[\sqrt{x}=x^\frac{1}{2}\]
Familiar with that?

Answer 7

yes

Answer 8

Then the next thing to see is that
\[\log(ab)=\log(a)+\log(b)\]
So you can rewrite the expression in the following way:
\[\log_b(\frac{x^\frac{1}{2}}{z^4}y^3)=\log_b(\frac{x^\frac{1}{2}}{z^4})+\log_b(y^3)\]

Answer 9

Okay to there?

Answer 10

it's that it?

Answer 11

No, not yet

Answer 12

Then you have to recognize the following:
\[\log(\frac{a}{b})=\log(a)-\log(b)\]
So you get
\[\log_b(\frac{x^\frac{1}{2}}{z^4})+\log_b(y^3)=\log_b(x^\frac{1}{2})-\log_b(z^4)+\log_b(y^3)\]

Answer 13

Next, we have to deal with the powers, so we use the following property:
\[\log(a^b)=b \log(a)\]
\[\log_b(x^\frac{1}{2})-\log_b(z^4)+\log_b(y^3)=\frac{1}{2}\log_b(x)-4\log_b(z)+3\log_b(y)\]

Answer 14

okay

Answer 15

so that's it? I will need to practice a lot to get that through my head.
lot of steps but it's worth it.

Answer 16

Yep, as long as you use one property at a time, it should work out.

Answer 17

another question

Answer 18

kk

Answer 19

Condense the expression to the logarithm of a single quantity.
1/3[lnx+3ln(x-2)]-ln(x+1)

Answer 20

You're going the other way with this one.
\[\frac{1}{3}(\ln(x)+3\ln(x-2))-\ln(x+1)\]
Like that?

Answer 21

first x don't need apprentices or maybe it don't matter

Answer 22

Shouldn't matter.

Answer 23

Alright, just like always, we're going to start with the innermost set of parentheses.

Answer 24

okay

Answer 25

Let's start with the part that says 3ln(x-2)
Use the property \[b \ln(a)=\ln(a^b)\]
We get
\[3\ln(x-2)=\ln((x-2)^3)\]

Answer 26

Next, we look at this part:
\[\ln(x)+\ln((x-2)^3)\]
We use the property
\[\ln(a)+\ln(b)=\ln(ab)\]
And we get
\[\ln(x)+\ln((x-2)^3)=\ln(x(x-2)^3)\]

Answer 27

So, basically, those two logarithms became one larger logarithm.

Answer 28

Here is where we're at now:
\[\frac{1}{3}(\ln(x(x-2)^3)-\ln(x+1)\]

Answer 29

Now we use the property
\[b \ln(a)=\ln(a^b)\]

We get
\[\ln((x(x-2)^3)^\frac{1}{3})-\ln(x+1)\]

Answer 30

big steps

Answer 31

Then, finally, we use the property\[\ln(a)-\ln(b)=\ln(\frac{a}{b})\]
\[\ln(\frac{(x(x-2)^3)^\frac{1}{3}}{x+1})\]

Answer 32

Yeah, need me to go through any of them again?

Answer 33

no I have a next problem give me a minute

Answer 34

approximate the following logarithms.
a) logb9

b)logb4/9

Answer 35

hello?

Answer 36

The given are logb2=0.3562, logb3=0.5646, log5=0.8271