Mastering Logarithms: Simplifying Complex Expressions with Multiple Logs

[rashida564]

Homework Statement

can we put
3log2(x)-4log(y)+log2(5)
in one logarithm
it try in all the ways but i can't find the solution .

Homework Equations

loga(b)=logx(b)/logx(a)
log(b*a)=log(b)+log(a)

The Attempt at a Solution

log2(5x^3)-log(y^4)
log2(5x^3)-log2(y^4)/log2(10)

 

[fresh_42]

Homework Statement

can we put
3log2(x)-4log(y)+log2(5)
in one logarithm
it try in all the ways but i can't find the solution .

Homework Equations

loga(b)=logx(b)/logx(a)
log(b*a)=log(b)+log(a)

The Attempt at a Solution

log2(5x^3)-log(y^4)
log2(5x^3)-log2(y^4)/log2(10)

There are also formulas for $\log_2 a - \log_2 b$ and $\log_2 a^c$ which you need here.

 

[rashida564]

i don't know that i should do

 

[fresh_42]

Well you have

log2(5x^3)-log(y^4)
log2(5x^3)-log2(y^4)/log2(10)

which I read as $\log_2 5x^3 - \log_{10} y^4 = \log_2 5x^3 - \frac{1}{\log_2 10}\log_2 y^4$.
Now you can use $c \cdot \log_2 a = \log_2 a^c$ and $\log_2 a - \log_2 b = \log_2 \frac{a}{b}$ to write all in a single $\log_2$ expression. (Of course with a constant $c=\log_2 10$.)

 

[rashida564]

log2(5x^3/((log2y^4)^(1/log2(10))))
then who i can write it as a single log i see three logs

 

[fresh_42]

log2(5x^3/((log2y^4)^(1/log2(10))))
then who i can write it as a single log i see three logs

You cannot get rid of the constant $\log_2 10$ if you are dealing with two different basis. Are you sure they are meant to be different?
And you have one $\log_2$ too many in the application of the formulas.

 

[rashida564]

sory for that

 

[rashida564]

sorry*