Solve the equation algebraically and check graphically.
$6300=300\left(10^x\right)$
$x \approx$ $\square$ (Round to three decimal places as needed.)
Answer (2)
Divide both sides of the equation by 300: 21 = 1 0 x
Take the logarithm base 10 of both sides: lo g 10 ( 21 ) = x
Calculate the value of lo g 10 ( 21 ) : x ≈ 1.322219
Round to three decimal places: x ≈ 1.322
Explanation
Problem Analysis We are given the equation 6300 = 300 ( 1 0 x ) and asked to solve for x , rounding to three decimal places.
Isolating the Exponential Term First, we divide both sides of the equation by 300 to isolate the exponential term: 300 6300 = 300 300 ( 1 0 x ) 21 = 1 0 x
Applying Logarithms Next, we take the logarithm base 10 of both sides of the equation: lo g 10 ( 21 ) = lo g 10 ( 1 0 x ) Using the property of logarithms that lo g b ( b x ) = x , we have: lo g 10 ( 21 ) = x
Calculating the Value of x Now, we calculate the value of lo g 10 ( 21 ) . The result of this operation is approximately 1.3222192947339193. Rounding this value to three decimal places, we get x ≈ 1.322 .
Final Answer Therefore, the solution to the equation 6300 = 300 ( 1 0 x ) , rounded to three decimal places, is x ≈ 1.322 .
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. Suppose you invest $300 in an account that pays an annual interest rate such that the investment grows to $6300 after some years. The equation 6300 = 300 ( 1 0 x ) can be used to find the number of years ( x ) it takes for the investment to reach the target amount, assuming the interest is compounded in a specific way.
We solved the equation 6300 = 300 ( 1 0 x ) by isolating the exponential term and taking logarithms, finding that x ≈ 1.322 rounded to three decimal places. A graphical check can also confirm this result by plotting the two sides of the equation. Thus, the final answer is x ≈ 1.322 .
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