Help please really stuck with logarithms :(?

Question

I really need help doing different functions with logs. I just read these 2 questions and I actually dont even know where to start so any help would be greatly appreciated.

The first one is

Solve
log*small 2* (2x + 1) - log*small 2* (x) = 2

And the other is
Given that log*small 5* (x) = a and
log*small 5* (y) = b
Find in terms a and b,

Log*small 5* (x*squared* divided by y)
and
Log (25x*square root y*)

All the x's are letters and not multiply signs by the way

I'm really stuck!

Answer 1

To help you solve this, rely on the following definition and try to understand it. It is a definition that is just being glossed over by a lot of students but actually it is a very powerful tool in the basic understanding of the logarithms.

The LOG of a number is the EXPONENT to which the base (like, 10 or 5 or 2 or 8, etc) is raised.

Example. Using base ten (decimal) 10^2. the log of 10^2 is the exponent 2. for 10^3 is 3 and so on.

Now the solution of the first problem is
1. COMBINE or condense the logs of base 2 by logs of a quotient
-- log a/b = log a - lob b
so, your given can be also written as log (2x+1)/(x) = 2
I will make your small 2 as base 2,

2. Now, take the base 2 of each side. Hard to type here.
something (base2)^log (2x+1)/(x) = base 2^2
just becomes (2x+1)/(x) = 4
2x+1 = 4x
x = 1/2

The second one is an application of the definition.
base 5^log x = 5^a
so x = 5^a
Doing the same thing for y , you get y = 5^b
Both give expressions in a and b.

You actually have more than 2 questions, (you lied), he he

Anyway, the next 2 questions are EXPANSION questions. Compare with COMBINING questions at the top.

log (base5) x^2/y = log (base5) x^2 - log (base5) y, application of log of quotient

log (base 10) 25xy^1/2 = log 25 + log x + 1/2 log y, application of log of product and powers(exponent)

Note: log (base 10) is just log

Hope this helps.

Answer 2

OK so 'small2' or 'small5'= the 'base' finally deciphired that.
Logarithims are exponents or powers of. So 10000(=10^4) has a log(small10!)=4 and 8(=2^3) has log(small2)=3. Indices of numbers are additive for multiplication and subtractive for division. That's why Napier devised them in the first place- to reduce lengthy multiplication and division to simple addition and subtraction using his famous tables. But you don't want all that explanation stuff! Here's the answers...
1) log(2x+1)-log(x)=2 becomes log(2x+1)/x=2
Now in the base 2 log system, by definition log2=1 (since 2 to the power of 1=2)
So we get (2x+1)/x=1 or 2x+1=x so x=-1
2)x=5^a and y=5^b (by defn) so xy=5^(a+b)
3) do not know what you are asking here!-sorry!

Answer 3

1. remember that log a - log b = log (a/b)
Then log base 2 (2x+1/x) = 2
then 2x+1 / x = 2^2 =4
then 2x + 1 =4x
then 1 = 2x
then x=1/2

2. log (base 5) X = A
then 5^A = X
then A x log 5 = log X
then A = log X / log 5

Answer 4

logb(c)= a means b^a = c

*logb means log to the base b*

log2 ([2x+1]/x) = 2

2^2 = (2x+1)/x

4x = 2x+1

2x = 1

x=0.5

Q2)

log5(x) = a , log5(y) = b

x = 5^a , y = 5^b

Other than that no idea sorry