Math 41 Sections 6.6, 6.7, and 6.8

Math 41 Sections 6.6, 6.7, and 6.8 - Math 41 6.6...

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Math 41 6.6 Logarithmic and Exponential Functions - y = log a x is equivalent to x = a y a > 0, a ≠ 1 - If log a M = log a N, then M = N. M,N, and a are positive and a ≠ 1. Example 1: Solving a Logarithmic Function Solve: 2 log 5 x = log 5 9 Solution: The domain of the variable in this equation is x > 0. Because each logarithm is to the same base, 5, we can obtain an exact solution as follows. 2 log 5 x = log 5 9 Log 5 x 2 = log 5 9 X 2 = 9 X = 3 Can’t be -3 since x > 0. Example 2: Solving a Logarithmic Function Solve: ln x + ln(x – 4) = ln(x + 6) Solution: The domain of the variable requires that x > 0, x – 4 > 0 and x + 6 > 0. As a result, the domain of the variable here is x > 4. We begin the solution using the log of a product property. Ln x + ln(x – 4) = ln(x + 6) Ln[x(x – 4)] = ln(x + 6) X(x – 4) = (x + 6) X 2 – 4x = x + 6 X 2 – 5x – 6 = 0 (x – 6)(x + 1) = 0 X = 6 or x = -1 Can’t be -1 since x > 4. - If a u = a v , then u = v a > 0, a ≠ 1 Example 3: Solving an Exponential Function Solve: 2 x = 5 Solution: Since 5 cannot be written as an integral power of 2, we write the exponential equation as the equivalent logarithmic equation. 2 x = 5
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Math 41 X = log x 5 X = ln5 / ln2 Change of base = 2.322
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Math 41 6.7 Compound Interest - the total amount borrowed from a bank, person, etc is known as the principal. The rate of interest , expressed as a percent, is the amount charged for the use of the principal for a given period of time. Simple Interest Formula I = Prt - When the interest due at the end of a payment period is added to the principal so that the interest computed at the end of the next payment period is based on this new principal amount, the interest is said to have been compounded . Compound interest is interest paid on principal and previously earned interest. Example 1: Computing Compound Interest A credit union pays interest of 8% per annum compounded quarterly on a certain savings plan. If $1000 is deposited in such a plan and the interest is left to accumulate, how much is in the account after 1 year? Solution: I = Prt I = (1000)(.08)(1/4) = $20 The new principal is 1020. At the end of the second quarter, the interest earned is I = (1020)(.08)(1/4) = $20.40 The new principle is 1040.40.
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