$7^{2 n}>52^{4 n+3}$
Answer (1)
Take the natural logarithm of both sides: (4n+3) \ln(52)"> 2 n ln ( 7 ) > ( 4 n + 3 ) ln ( 52 ) .
Distribute and isolate n : 3 \ln(52)"> n ( 2 ln ( 7 ) − 4 ln ( 52 )) > 3 ln ( 52 ) .
Divide by ( 2 ln ( 7 ) − 4 ln ( 52 )) , remembering to flip the inequality sign because this term is negative: n < 2 l n ( 7 ) − 4 l n ( 52 ) 3 l n ( 52 ) .
Simplify to find the approximate bound for n : n < − 0.995 .
n < − 0.995
Explanation
Analyzing the Inequality We are given the inequality 52^{4 n+3}"> 7 2 n > 5 2 4 n + 3 . Our goal is to find the values of n that satisfy this inequality. To do this, we will take the logarithm of both sides, simplify, and isolate n .
Applying Logarithms Taking the natural logarithm (ln) of both sides of the inequality 52^{4 n+3}"> 7 2 n > 5 2 4 n + 3 , we get:
\ln(52^{4n+3})"> ln ( 7 2 n ) > ln ( 5 2 4 n + 3 )
Using the logarithm power rule, which states that ln ( a b ) = b ln ( a ) , we can rewrite the inequality as:
(4n+3) \ln(52)"> 2 n ln ( 7 ) > ( 4 n + 3 ) ln ( 52 )
Isolating n Now, we distribute ln ( 52 ) on the right side:
4n \ln(52) + 3 \ln(52)"> 2 n ln ( 7 ) > 4 n ln ( 52 ) + 3 ln ( 52 )
Next, we want to isolate n . We move all terms containing n to one side of the inequality:
3 \ln(52)"> 2 n ln ( 7 ) − 4 n ln ( 52 ) > 3 ln ( 52 )
Factor out n :
3 \ln(52)"> n ( 2 ln ( 7 ) − 4 ln ( 52 )) > 3 ln ( 52 )
Solving for n Now, we divide both sides by ( 2 ln ( 7 ) − 4 ln ( 52 )) to solve for n . However, we must be careful about the sign of this expression. Let's determine the sign:
2 ln ( 7 ) − 4 ln ( 52 ) = 2 ( ln ( 7 ) − 2 ln ( 52 )) = 2 ( ln ( 7 ) − ln ( 5 2 2 )) = 2 ( ln ( 7 ) − ln ( 2704 ))
Since 7 < 2704 , we have ln ( 7 ) < ln ( 2704 ) , so ln ( 7 ) − ln ( 2704 ) < 0 . Thus, 2 ln ( 7 ) − 4 ln ( 52 ) < 0 . Therefore, when we divide by this negative number, we must reverse the inequality sign:
n < 2 ln ( 7 ) − 4 ln ( 52 ) 3 ln ( 52 )
Calculating the Bound We can simplify the expression further:
n < 2 ( ln ( 7 ) − 2 ln ( 52 )) 3 ln ( 52 )
Using a calculator, we find that ln ( 7 ) ≈ 1.9459 and ln ( 52 ) ≈ 3.9512 . Plugging these values in, we get:
n < 2 ( 1.9459 − 2 ( 3.9512 )) 3 ( 3.9512 ) = 2 ( 1.9459 − 7.9024 ) 11.8536 = 2 ( − 5.9565 ) 11.8536 = − 11.913 11.8536 ≈ − 0.995
Therefore, n < − 0.995 .
Final Answer Thus, the solution to the inequality 52^{4 n+3}"> 7 2 n > 5 2 4 n + 3 is approximately n < − 0.995 .
Examples
Understanding exponential inequalities is crucial in various fields, such as finance and computer science. For instance, in finance, it helps in modeling the decay of investments or the growth of debt under certain interest rates. In computer science, it can be used to analyze the efficiency of algorithms, where the execution time might grow exponentially with the input size. By solving such inequalities, one can determine the conditions under which an investment remains profitable or an algorithm remains efficient.
