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(And we only make this definition in the case where .
And, to solve an equation, I have to get the variable by itself on one side of the "equals" sign; to isolate the variable, I have to "undo" whatever has been done to the variable.
In this case, the variable has been put in the exponent.
Similarly, the exponentiation is defined as the repetition of multiplication.
For example, writing out can get boring fast, so we define the exponential function to express this in a much more compact form so that the preceeding example can be written as (read 3 to the 5th or 3 to the 5 power).
Exponentiation is an arithmetic operation, just like addition, multiplication, etc.
It is the second operation performed if a equation has parentheses or the first one performed when there is no parentheses.
As with most problems in basic algebra, solving large exponents requires factoring.
If you factor the exponent down until all the factors are prime numbers – a process called prime factorization – you can then apply the power rule of exponents to solve the problem.
The abbreviation is pronounced "ell-enn" and written with a lower-case "L" followed by a lower-case "N". Since science uses the natural log so much, and since it is one of the two logs that calculators can evaluate, I tend to take the natural log of both sides when solving exponential equations.
This is not (generally) required, but is often more useful than other options.