Unformatted text preview: MTH405 Wk 4 Lect 1 Change of Base Formula
Let a, b and x be positive real numbers such that a 6= 1 and b 6= 1. Then
loga x = ln x
.
ln a Exercise. 1. Use natural logarithms to evaluate log3 5.
2. Use natural logarithms to evaluate log6 2. Solving Exponential and Logarithmic equations
(Properties of Exponential and Logarithmic equations) Let a be a
positive real number such that a 6= 1, and let x and y be real numbers. Then the
following properties are true. Theorem. 1. ax = ay if, and only if, x = y .
(a) loga x = loga y if, and only if, x = y (x > 0, y > 0).
(b) loga (ax ) = x and ln(ex ) = x.
(c) a(loga x) = x and e(ln x) = x.
Exercise. Use the properties above to solve each equation. 1. 4x+2 = 64.
2. ln(2x − 3) = ln 11.
3. 2 log x = log 16.
4. 1
2 log x = log 6. 5. logx 25 = 2.
6. x log2 8 = 6.
7. log4 x = 2.
8. log5 x = 3.
1 Solving exponential equations
To solve an exponential equation, rst isolate the exponential expression. Then
take the logarithm of each side of the equation and solve for the variable.
Exercise. Solve each exponential expression. 1. 2x = 7.
2. 4x−3 = 9.
3. 2ex = 10.
4. 5 + ex+1 = 20. Solving Logarithmic Equations
To solve a logarithmic equation, rst isolate the logarithmic expression. Then
exponentiate each side of the equation and solve for the variable.
Exercise. Solve each logarithmic equation. 1. 2 log4 x = 5.
2. 1
4 log2 x = 12 . 3. 3 log10 x = 6.
4. log3 2x − log3 (x − 3) = 1.
Exercise. A deposit of $5000 is placed in a savings account for 2 years. The
interest on the account is compounded continuously. At the end of 2 years, the
balance in the account is $5416.44. What is the annual interest rate for this
account?
r n
) where P =initial amount, A =nishing
( Hint: Amount A = P (1 + 100
amount, r =annual interest rate, n =no. of periods of the interest ) 2 ...
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 Spring '16
 Alveen Chand
 Logarithmic Equations, Natural logarithm, Logarithm, logarithmic equation

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