Solve the equation without using your calculator. [tex]10^n=100[/tex]
Answer (2)
Rewrite 100 as a power of 10: 100 = 1 0 2 .
Express the equation as 1 0 n = 1 0 2 .
Equate the exponents: n = 2 .
The solution to the equation is 2 .
Explanation
Understanding the Problem We are given the equation 1 0 n = 100 and we need to find the value of n .
Rewriting 100 as a power of 10 We can rewrite 100 as a power of 10. We know that 100 = 10 × 10 = 1 0 2 .
Expressing the equation with the same base Now we can rewrite the original equation as 1 0 n = 1 0 2 .
Equating the exponents Since the bases are the same, we can equate the exponents. Therefore, n = 2 .
Examples
Exponents are used to calculate compound interest. For example, if you invest 100 a t anann u a l in t eres t r a t eo f 5 n$ years is 100 ( 1.05 ) n . Solving exponential equations helps determine how long it takes for an investment to reach a certain value. In this case, understanding exponents allows you to calculate the growth of your investment over time and plan accordingly.
To solve the equation 1 0 n = 100 , we rewrite 100 as 1 0 2 . After expressing both sides with the same base, we equate the exponents to find that n = 2 . Thus, the solution is 2 .
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