Prove that 3 + 2√5 is irrational
1 answer:
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Answer:
b. e^9.45 = x
see last example and this explains whole numbers and decimals.
Step-by-step explanation:
Another example we can Solve 100=20e^2t .
Solution
100 = 20e^2t
5 = 20e ^2t
in 5 = 2t
Therefore t = in5/ 2
Step 1 was ; Divide by the coefficient of the power
Step 2 was ; Take ln of both sides. Use the fact that ln(x) and ex are inverse functions
Step 3 was; Divide by the coefficient of t
Another example;
Solve e^2x−e^x = 56 .
Solution
Analysis
When we plan to use factoring to solve a problem, we always get zero on one side of the equation, because zero has the unique property that when a product is zero, one or both of the factors must be zero. We reject the equation e^x=−7 because a positive number never equals a negative number. The solution ln(−7) is not a real number, and in the real number system this solution is rejected as an extraneous solution.
Another example is;
Solve e^2x=e^x+2 .
Answer
Q&A: Does every logarithmic equation have a solution?
No. Keep in mind that we can only apply the logarithm to a positive number. Always check for extraneous solutions.
Last example determines decimals ;
Solve lnx =3 .
Solution
lnx^x=3=e^3
Use the definition of the natural logarithm
Graph below represents the graph of the equation. On the graph, the x-coordinate of the point at which the two graphs intersect is close to 20 . In other words e^3≈20 . A calculator gives a better approximation: e^3≈20.0855 .
The graph below represents the graph of the equation. On the graph, the x-coordinate of the point at which the two graphs intersect is close to 20 . In other words e^3≈20 . A calculator gives a better approximation: e^3≈20.0855 .
It shows values of graphs of y=lnx and y=3 cross at the point (e^3,3) , which is approximately (20.0855,3) .
See graph below.
