Can someone help me understand this Logarithm stuff, please?

Question

Here are some problem examples:

A. log2x 16 = -2

B. log4 (2x) > -1/2

C. logx+4 27 = 3

D. logx 1000 = 3

I do not get how this works. Can someone explain, please? Thanks so much!

Answer 1

A
log2x 16 =-2
2x ^(-2) = 16
2x = 16^(-1/2)
2x = 1/4
x = 1/8

B
log4 (2x) > -1/2
4^(-1/2) < 2x
1/2<2x
x>1/4

C
logx+4 27 = 3
(x+4)^3 = 27
x+4 = 27^(1/3)
x+4 = 3
x = -1

D
logx 1000 = 3
x^3 = 1000
x = 1000^(1/3)
x =10

Answer 2

the number folowing the log is called the base. the next number is what you are getting the log of and the number on the other side of the equal sign is the log of the number. it is simple the pwewr that you must raise the base to to yield the given number.

I find it helpful to remember a simple example:
log10 (100) = 2 this means 10^2 =100 which we know is correct. using this for \mat we caneasily solve any of this type of problem.

A. log2x 16 =-2
2x^-2 =16 or 1/(2x)^2 = 16
1 =( 4x^2)16
4x^2=1/16
x^ 2= 1/64
x= 1/8

b. Log4 (2x) =-1/2
4^-1/2 =2x
The minus 1/2 power means 1 /sqrt
1/sqrt 4 = 2x
1 = (sqrt 4)2x = 4x
x = 1/4

C. logx+4(27) =3
27 = (x+4) ^3
or take the cubed root of each side
3 = x + 4
x= -1

D. logx 1000 =3
1000 = x^3
taking cubed root:
10 = x

Answer 3

Just remember that a log is an exponent. Your basic rules are as follows:
Log (AB) = Log(A) + Log(B)
Log(A/B) = Log(A) - Log(B)
Log(A ^ B) = B * log(A)
Log (10 ^x) = x
10 ^ (Log(x)) = x
Here is how I always remember logs. I know that Log(1000) = 3. Since it is to base 10, you ask yourself 10 ^ ? = 1000. The answer is 3, since 10^3 = 1000.

A) Using my example above, log base b (x) = y is equivalent to
b ^ (y) = x. So rewrite this as an exponent using this example.
Your base is 2x, so therefore
2x ^ (-2) = 16
Now take the negative square root of both sides (-1/2 power), to get 2x by itself.
2x = 16 ^ (-1/2)
2x = 1 / (16 ^ 1/2)
2x = 1 / 4
Now divide by 2, giving you
x = 1/8.

Answer 4

Problem a, which i will help you with (then its up to you to do your own homework) converts to exponential form as:

2x^(-2)=16

now just solve x by first distributing the power:

1/(4x^2)=16

1/4=16x^2

1/64=x^2
sqrt(1/64)=x
x=1/8

Answer 5

as a results of fact each and every of the themes have the comparable log base it turns into beside the point. Logrithms could be simplified in accordance to the operation it relatively is meant to be completed including- multiply the two EX: Log(3)7+Log(3)x = Log(3)7x Subtracting - divide the numbers EX Log(b)20-Log(b)7x = Log(b) 20/7x the coefficents interior the final situation are elevated or divided in accordance to the sign aswell. do purely each and every situation in steps and that they improve into common one million. log(2)5+log(2)x-log(2)10 log(2)5x-log10 log(2)5x/10 then simplify the fraction log(2)x/2