Question

Can someone please solve: Solve log ( 2x + 1 ) – log 6 = 2 for x. Please explain how you arrived at your answer.

Answer 1

x=299.5

Explanation:

I will convert log(6) to its decimal form

`log(2x+1)-log(6)=2\\\\log(2x+1)-0.778151=2`

Step 1: Add 0.778151 to both sides.

`log(2x+1)-0.778151+0.778151=2+0.778151\\\\log(2x+1)=2.778151`

Step 2: Solve Logarithm.

`log(2x+1)=2.778151\\\\10^(log(2x+1))=10^(102.778151)` Take the exponent of both sides.

`2x+1=102.778151\\\\2x+1=599.999654\\\\2x+1-1=599.999654-1`

Subtract 1 from both sides.

`2x=598.999654\\\\\\(2x)/(2)=(598.999654)/(2)`

Divide both sides by 2.

`x=299.5`

Answer 2

599/2

Explanation:

`\log(2x+1)-\log(6)=2`

I'm going to use quotient rule which says:
`\log((q)/(p))=\log(q)-\log(p)`:

`\log((2x+1)/(6))=2`

Now we are going write this in equivalent expontial form. That is,
`\log_b(a)=y` implies
`b^y=a`.

Writing in exponetial form:

`10^2=(2x+1)/(6)`

`100=(2x+1)/(6)`

Multiply both sides by 6:

`600=2x+1`

Subtract 1 on both sides:

`599=2x`

Divide both sides by 2:

`(599)/(2)=x`

When it comes to logarithms, you should check your solution(s).

`\log(2 \cdot (599)/(2)+1)-\log(6)=2`

`\log(599+1)-\log(6)=2`

`\log(600)-\log(6)=2`

`\log((600)/(6))=2`

`\log(100)=2`

`2=2`

The solution checks out.