Write as a single logarithm : log 3x - 2log x + log(x +3)

Question

Write the following separate logarithm as a single logarithm : log 3x - 2log x + log(x +3)

All the log is to the base 2.

Answer 1

Hi Suama!

 

 

Since all the log terms are the same base, the terms can be further simplified using log properties.

 

Log properties:

 

1.  log_bb=1

2.  log_bb^x=x

3.  log_b(x^r)= r log_b(x)

4.  log_b(xy)=log_b(x)+log_by

5. logb(x/y)=log_b(x) - log_b(y)

 

To simplify your problem:

 

                                                  log₂ 3x - 2log₂ x + log₂(x +3)

rule #3                                        log₂ 3x - log₂ x² + log₂(x +3)

rule #4                                        log₂ 3x - log₂ x²(x +3)

rule #5                                        log₂ [3x / x²(x +3)]

reduce/simplify                             log₂ [3/ x(x +3)]

 

The response can be left like that or the denominator can be multiplied out.  Depending on what your teacher requires.  I don't require my students to multiply out the denominator.

 

 

I hope this helps you with the process of simplifying logarithms.

THINK MATH!!

 

Answer 2

log (3x) - 2log(x) + log(x+3)

= log(3x) - log(x^2) + log(x+3)

= log((3x)(x+3)/(x^2))