$-7 \cdot 3^{0.25 x}=-10$
Which of the following is the solution of the equation?
Choose 1 answer:
(A) $x=4 \log _3\left(\frac{10}{7}\right)$
(B) $x=\frac{4 \log _3(7)}{\log _3 10}$
(C) $x=\frac{4 \log _3(10)}{\log _3(7)}$
(D) $x=4 \log _{\frac{10}{7}}(3)$
Answer (1)
Divide both sides by -7: 3 0.25 x = 7 10 .
Take the logarithm base 3 of both sides: 0.25 x = lo g 3 ( 7 10 ) .
Multiply both sides by 4: x = 4 lo g 3 ( 7 10 ) .
The solution is x = 4 lo g 3 ( 7 10 )
Explanation
Problem Analysis We are given the equation − 7 c d o t 3 0.25 x = − 10 and asked to find the value of x .
Isolating the Exponential Term First, we divide both sides of the equation by − 7 to isolate the exponential term: 3 0.25 x = − 7 − 10 = 7 10 .
Taking the Logarithm Next, we take the logarithm base 3 of both sides of the equation: lo g 3 ( 3 0.25 x ) = lo g 3 ( 7 10 ) .
Simplifying the Logarithm Using the property of logarithms that lo g b ( a c ) = c lo g b ( a ) , we simplify the left side: 0.25 x = lo g 3 ( 7 10 ) .
Solving for x Now, we multiply both sides by 4 to solve for x : x = 4 lo g 3 ( 7 10 ) .
Final Answer Comparing our result with the given options, we see that it matches option (A).
Examples
Exponential equations are used in various fields such as finance, biology, and physics. For example, in finance, they are used to model compound interest. Imagine you invest a certain amount of money in an account that compounds interest. The exponential equation helps you calculate how long it will take for your investment to reach a specific target amount, considering the interest rate and compounding frequency. This understanding is crucial for making informed financial decisions and planning for the future.
