SOLUTION: Find the solution sets for the following: In (3x) = 6

Question 92161: Find the solution sets for the following:
In (3x) = 6
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
Given:
.
ln (3x) = 6
.
By the rules of logarithms, the log of a product is equal to the sum of the logs of each
of the terms in the product. Therefore, ln(3*x) = ln(3) + ln(x). Substitute this into the
given equation and it becomes:
.
ln(3) + ln(x) = 6
.
Subtract ln(3) from both sides to get rid of that term on the left side. After this
subtraction the equation is:
.
ln(x) = 6 - ln(3)
.
But ln(3) is just a number that you can get from a scientific calculator. Enter 3 and then
press the ln key. You will find that ln(3) is equal to 1.098612289. Replace ln(3) in
the equation with that number and you get:
.
ln(x) = 6 - 1.098612289
.
Perform the subtraction on the right side of the equation and the equation reduces to:
.
ln(x) = 4.901387711
.
Next convert this logarithmic equation into it equivalent exponential form using the following
rule of logarithms:
.
is equivalent to
.
And in this problem the base, b, of the natural logarithms is the number e (which is
2.718281828).
.
So converting to exponential form results in an equation transformation of:
.
converting to
.
and since e is approximately 2.718281828 this becomes:
.

.
Using a scientific calculator to raise 2.718281828 to the 4.90138771 power results in
an answer of:
.
x = 134.4762643
.
You also could have done the problem by beginning with the conversion to exponential
form directly. That is, starting with:
.
ln(3x) = 6
.
immediately raise the base e to the sixth power and set it equal to 3x as follows:
.

.
Then divide both sides by 3 to find that:
.

.
Then you can raise e to the sixth power and get an answer for this
of 403.4287935 which you then divide by 3 to find that x = 134.4762645
.
The slight difference in the very last decimal place between this answer and the answer we
got using the first method involves just calculator round off in all the steps of the
first method. This second method is a little more accurate because the round off only occurs
in the very last calculation.
.
Hope this helps you to understand the problem a little better. The basis of answering
this problem is knowing how to convert from logarithmic form to exponential form and you
need to memorize that conversion. It gets used a lot in solving logarithmic equations.
.
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