I was working out a problem and reached an answer of 3ln |x|+ (1/x)+ ln |x+1|
The final given answer is (1/x) + ln |x^4 +x^3|
I see that the (1/x) carried over and I know how the x^3 got there, but I'm confused as to the x^4. Can someone kindly explain this to me?
3 ln x = ln x^3
ln x^3 + ln|x+1| = ln [x^3(x+1)]
= ln [x^4 + x^3]
Thanks Damon! My math skills are sadly not where they should be.
You are welcome.
To understand how the x^4 term appears in the final answer, let's break down the given expression you reached: 3ln |x| + (1/x) + ln |x+1|.
First, let's focus on the 3ln |x| term. When you have the natural logarithm of absolute value of x, ln |x|, and you multiply it with a constant (in this case, 3), you can use the property of logarithms to bring the exponent inside as a coefficient. So, 3ln |x| can be simplified as ln |x|^3.
Next, let's examine the (1/x) term. If we multiply (1/x) by x, we get 1. However, if we want to keep the original expression unchanged, we need to multiply by x/x instead, which is equivalent to 1. So, (1/x) can be written as (x/x^2) or (x^-1).
Lastly, we have ln |x+1|. Now, this term gives rise to the appearance of the x^4 in the final answer. Suppose we want to find a power of x that is higher than x^3. We need to manipulate the expression to include higher powers of x. One way to do this is by using a power rule for logarithms, which states that ln (ab) = ln a + ln b.
In this case, we can rewrite x+1 as (x+1)^1. Now, if we rewrite ln |x+1| using the power rule, we get ln (|x|+1). To keep the original expression unchanged, we need to subtract ln |x| (which is equivalent to adding its negative). Therefore, ln (|x|+1) is equal to ln |x| + ln (1+|x|).
At this point, we have ln |x|^3 + (x^-1) + ln |x| + ln (1+|x|). To simplify this further, we can combine the logarithmic terms by adding their coefficients. So, ln |x|^3 + ln |x| + ln (1+|x|) gives us ln (|x|^3 * |x| * (1+|x|)), which simplifies to ln |x^4 + x^3 + x^2 + x|.
To match the given answer, we need to factor out an x from inside the absolute value: ln |x^4 + x^3 + x^2 + x| = ln |x * (x^3 + x^2 + x + 1)|.
And finally, we can rewrite (x^3 + x^2 + x + 1) as x^4 + x^3 by multiplying both sides by x.
Therefore, the given answer, (1/x) + ln |x^4 + x^3|, is equivalent to the expression you reached, 3ln |x| + (1/x) + ln |x+1|.