In increasing order, rank:
3^ln 2, 2^3, 2^ln 3
*do not use a calculator, and explain your choices
Obviously, 2^3=8
So, instinctively, I entered the other values into my calculator just to rank the quantities using decimal places. However, in my calculator, 3^ln 2 and 2^ln 3 both yield: 2.1414886064.
So, can someone explain this problem to me please!
2^3
take ln and get ln(2^3) = 3 ln 2
3^ln 2
take ln and get ln (3^ln 2) = ln 2 ln 3
2^ln 3 also gives ln 3 ln2
and ln 3 <3
To rank the quantities without using a calculator, we need to rely on some properties of exponential and logarithmic functions.
Let's start by comparing 3^ln 2 and 2^ln 3.
We know that ln 2 is the natural logarithm of 2, and ln 3 is the natural logarithm of 3. The natural logarithm is a monotonic function, which means it preserves the order of its inputs. Therefore, if we compare ln 2 and ln 3, we can determine the order of 3^ln 2 and 2^ln 3.
ln 2 is approximately 0.6931, and ln 3 is approximately 1.0986. So, ln 3 is greater than ln 2.
Now, let's compare 3^ln 2 and 2^3.
We can rewrite 3^ln 2 as (e^(ln 3))^ln 2. Using exponent rules, this simplifies to e^(ln 3 * ln 2).
Similarly, 2^3 can be written as e^(ln 2 * 3).
Comparing the exponents, we can determine the order by comparing ln 3 * ln 2 and ln 2 * 3.
ln 3 * ln 2 is approximately 0.7588, and ln 2 * 3 is approximately 2.0794. Therefore, ln 2 * 3 is greater than ln 3 * ln 2.
Therefore, we can conclude that 3^ln 2 is between 2^3 and 2^ln 3.
In increasing order, the rank is:
2^3 < 3^ln 2 < 2^ln 3.