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Question: How do you simplify \(\log \left( {{10}^{-7}} \right)=x\) ?...

How do you simplify log⁡(10−7)=x\log \left( {{10}^{-7}} \right)=x ?

Explanation

Solution

There are numerous properties and formulas of logarithms. For solving these types of problems, we have to keep one property of logarithms in mind, which is
log⁡ab=blog⁡a\log {{a}^{b}}=b\log a
Where bb can be any positive or negative number. Therefore, we simplify log⁡(10−7)\log \left( {{10}^{-7}} \right) accordingly as −7log⁡10-7\log 10 . log⁡10\log 10 evaluating to 1, the expression gets finally simplified to −7-7 .

Complete step by step answer:
The given equation is
log⁡(10−7)=x....equation1\log \left( {{10}^{-7}} \right)=x....equation1
Now, we apply the basic formula or property of logarithms which states that,
log⁡ab=blog⁡a\log {{a}^{b}}=b\log a
Where, bb can be any number, positive or negative and the logarithm can be natural or common or of any other base. In our problem, after analysing, we find that a=10a=10 and b=−7b=-7 . Therefore, applying this property to the LHS of equation1equation1 , we get
⇒−7log⁡10=x\Rightarrow -7\log 10=x
As the logarithm given in the problem is not mentioned, so we consider it as a common logarithm, that is, logarithm with base 1010 . Then log⁡10\log 10 becomes simply 11 . Thus,
⇒−7×1=x\Rightarrow -7\times 1=x
⇒−7=x\Rightarrow -7=x
Therefore, we can conclude that the solution of the given equation is x=−7x=-7 .

Note: Students must be careful while applying the properties of logarithms and must not confuse the property used in this problem with some other. If the type of logarithm is not mentioned in the question, such as in this case, then we need to assume it according to the simplification of the problem and see which one simplifies the problem more. The negative terms usually create a mess during solving, so we must be careful with them. We should not take 1010 raised to power on both sides of the equation as this will not help in this problem.