Question 80372: I would like a solution to how it is done to evaluate each expression:
-2+ 1n e^3. I do know the answer is 1 (from the back of my text book's answers to the problem), but the log calculator on this web site had the answer as 0 because it was needed 0 divisions to get to 1. Please explain .
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Given the expression:
.
-2 + ln e^3
.
Simplify this expression.
.
You can just about do this in your head by applying two rules of logarithms. The first rule
is that if you take the logarithm of a quantity that with an exponent, you can make an equivalent
term by multiplying the logarithm of the quantity times the exponent. If you apply this
rule to the given expression, you convert the expression to:
.
-2 + 3*ln e
.
The second rule (actually a definition) is that you can convert a logarithm to exponential
form by raising the base to the exponent on the other side and setting that equal to
the quantity you are taking the logarithm of. Easier to do than to say. Let's use this
rule/definition to find ln e.
.
Let's set y equal to ln e. We know that the base of the natural logarithms is e. If
we raise that base to the exponent y it will equal the quantity that the ln function
is operating on. So
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ln x = y means (base)^y = x
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and since the (base) is e we can say e^y = x
.
Now notice that for this problem x is e. Substituting e for x gives us
.
e^y = e
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If you look at this carefully, you can see that to make the left side equal to the
right side, the exponent y has to be 1 ... making the equation e^1 = e.
.
So we now know that y = 1, but recall that we had said y = ln e. This tells us that
1 = ln e
.
Note - you can also do this on a scientific calculator. Fine the function key for e^x.
Enter 1 for x and then press the e^x key. You should get 2.718281828 as the value of
e^1 which is the same as e. Then press the ln key to take the natural logarithm of
2.718281828. You should get 1 as the answer. This tells you that the natural logarithm
of e (or 2.718281828) is 1 ... just as we found in the previous paragraph.
.
Anyhow, if we go back to the expression in the form:
.
-2 + 3*ln e
.
and we substitute 1 for ln e, we get:
.
-2 + 3*1 = -2 + 3 = +1
.
And that's the book answer.
.
Hope this helps you with your understanding of logarithms in general and natural logarithms
in particular.
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