Let a be a positive number such that a does not equal 1, let n be a real number, and let u and v be positive real numbers.

Logarithmic Rule 3: tex2html_wrap_inline62 .

Example 8: Simplify tex2html_wrap_inline64 .

Solution: tex2html_wrap_inline64 can be written tex2html_wrap_inline68 . In its present form there is nothing else we can do simplify the expression. Remember the rules of logarithms deal with products, quotients and powers, not sums or differences. Let's see if we can turn tex2html_wrap_inline70 into a product. We can. The expression tex2html_wrap_inline70 can be written in an equivalent form as the product of two factors: tex2html_wrap_inline74 . Now the expression tex2html_wrap_inline68 can therefore be written as tex2html_wrap_inline78 .

Is the initial expression tex2html_wrap_inline64 always equal to the final expression tex2html_wrap_inline78 ? No!

Note that the initial expression tex2html_wrap_inline64 is valid only when tex2html_wrap_inline86 . This means that if choose x > 6 or we choose x < 1, we can find the value of the initial expression.

You have to check/make sure that the number you take the logarithm of is positive. In our case, we must have that x2 - 7 x +6 is strictly positive.

If you would like to work some problems, click here.

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Author: Nancy Marcus

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