4-5 And 4-6 Quiz: Logarithmic Expressions And Equations

Explanation
The given equation is x = 9. This means that the value of x is equal to 9.

Explanation
The given equation has multiple values of x. However, the correct answer is x = 6 because it is the only value that is listed in the options provided. The other values (4, 16, and 9) are not listed as possible solutions, so they are not the correct answers.

Explanation
The general formula for growth and decay is given by the equation: N(t) = N0 * e^(kt), where N(t) is the number of bacteria at time t, N0 is the initial number of bacteria, e is the base of the natural logarithm, and k is the growth or decay constant. In this case, we know that the initial number of bacteria is 80 and the number of bacteria after 3 hours is 164. Plugging these values into the formula, we can solve for k.

Explanation
Jason invests $1000 in an account that pays 2.5% interest compounded continuously. Continuous compounding means that the interest is constantly being added to the account balance. To calculate the amount in the account at the end of 2 years, we can use the formula A = P * e^(rt), where A is the final amount, P is the principal (initial investment), e is the base of the natural logarithm, r is the interest rate, and t is the time in years. Plugging in the values, we get A = 1000 * e^(0.025 * 2) ≈ $1051.27.

Explanation
The value of the machinery depreciates at a rate of 20% per year. To find out how many years it will take for the value to depreciate to $15,000, we need to calculate the depreciation amount per year. The depreciation amount per year can be found by multiplying the initial value of the machinery ($60,000) by the depreciation rate (20% or 0.2). The depreciation amount per year is $12,000. To find out how many years it will take for the value to depreciate to $15,000, we divide the difference between the initial value and the desired value ($60,000 - $15,000 = $45,000) by the depreciation amount per year ($12,000). Therefore, it will take approximately 3.75 years for the value to depreciate to $15,000.

Explanation
The formula for calculating the amount in an account with continuous compounding is A = P * e^(rt), where A is the final amount, P is the principal amount, e is Euler's number (approximately 2.71828), r is the interest rate, and t is the time in years. Plugging in the values, we get A = 1200 * e^(0.035 * 2) = 1200 * e^(0.07) ≈ $1287.01. Therefore, the correct answer is $1287.01.

Explanation
The value of the machinery decreases by 20% each year. To find out how many years it takes for the value to depreciate to $15,000, we can set up an equation. Let x represent the number of years. The equation would be: $45,000 * (1 - 0.20)^x = $15,000. By solving this equation, we can find the value of x, which represents the number of years it takes for the machinery to depreciate to $15,000.