Which value makes the equation $3.62 \times 10^{\circ}=7$ true?
A. 36.200 .000
B. 36.200
C. 362.000
D. 3.620
Answer (1)
Divide both sides of the equation by 3.62: 1 0 ∘ = 3.62 7 .
Take the base-10 logarithm of both sides: ∘ = lo g 10 ( 3.62 7 ) .
Calculate the value of the logarithm: ∘ ≈ 0.2864 .
The value of ∘ that makes the equation true is approximately 0.2864.
Explanation
Understanding the Problem We are given the equation 3.62 × 1 0 ∘ = 7 and we need to find the value of ∘ that makes this equation true. The options provided are (A) 36.200 .000, (B) 36.200, (C) 362.000, and (D) 3.620. Our goal is to isolate 1 0 ∘ and then solve for ∘ .
Isolating the Exponential Term First, we divide both sides of the equation by 3.62 to isolate the term with the unknown exponent:
Dividing Both Sides 1 0 ∘ = 3.62 7
Taking the Logarithm Next, we take the base-10 logarithm of both sides of the equation to solve for ∘ . Using the property that lo g 10 ( 1 0 x ) = x , we have:
Simplifying the Equation lo g 10 ( 1 0 ∘ ) = lo g 10 ( 3.62 7 ) ∘ = lo g 10 ( 3.62 7 )
Calculating the Logarithm Now, we calculate the value of lo g 10 ( 3.62 7 ) . The result of this calculation is approximately 0.2864.
Comparing with Options Comparing the calculated value (approximately 0.2864) with the given options, we see that none of the options (A) 36.200 .000, (B) 36.200, (C) 362.000, (D) 3.620 are close to this value. However, there seems to be a misunderstanding in the options. The question is asking for the value of ∘ , not 1 0 ∘ . The value of ∘ is approximately 0.2864.
Final Answer Since none of the provided options match the calculated value of approximately 0.2864, it seems there might be an error in the question or the provided options. However, based on our calculations, the correct value for ∘ is approximately 0.2864.
Examples
Understanding exponential equations is crucial in various fields like finance and science. For instance, calculating the growth of an investment with compound interest involves solving equations similar to the one in this problem. If you invest P dollars at an annual interest rate r compounded n times per year, the amount A after t years is given by A = P ( 1 + n r ) n t . Solving for t or r would require using logarithms, just like in our problem. This skill is also essential in understanding radioactive decay, population growth, and many other real-world phenomena.
