Find x when: (2^x)(e^(3x+1))=10. Give your answer in the form (a + ln(b)) / (c + ln(d)) , where a,b,c,d are integers.
First notice x is currently written as a power of 2 and 3x + 1 is written as a power of e, so we are going to need to use some sort of log to start getting x on its own. You can take a clue from the question that we should use the natural logarithm (ln) on both sides of the equation:
ln( 2x * e(3x+1) ) = ln(10)
The next thing to notice is that on the left hand side, we have a product of two expressions in the log. So we can use the identity ln(ab) = ln(a) + ln(b) to write the above as:
ln(2x) + ln(e(3x+1)) = ln(10)
Then we can use the identity log(ak) = k * log(a) to write this as:
x * ln(2) + (3x+1) * ln(e) = ln(10)
As ln(e) is 1:
x * ln(2) + 3x + 1 = ln(10)
Next we must collect the x terms:
(ln(2) + 3)x = ln(10) - 1
Finally divide through by (ln(2) + 3) to get:
x = (-1 + ln(10)) / (ln(2) + 3) which is in the form asked for.
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