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Math 1014: Precalculus with Transcendentals Ch. 4: Exponential and Logarithmic Functions Sec. 4.4: Exponential and Logarithmic Equations I. Exponential Equations A. Definition An exponential equation is an equation containing a variable in an exponent. B. Solving Exponential Equations by Expressing Each Side as a Power of the Same Base If b M = b N , then M = N. 1. Rewrite the equation in the form 2. Set M=N. bM = bN . 3. Solve for the variable. C. Examples Solve for x: 1. 3x = 81 2. 2 4 x−1 = 8 D. Using Logarithms to Solve Exponential Equations 1. Isolate the exponential expression. 10. Take the natural logarithm on both sides of the equation for bases other than 10. 2. Take the common logarithm on both sides of the equation for base 3. Simplify using one of the following properties: ln(b x ) = x ln(b) or ln(e x ) = x or log (10 x ) = x 4. Solve for the variable. E. Examples Solve for x: 1. 10 x = 8.07 2. 9e x = 107 II. 3. 5 x−3 = 137 4. e4 x − 3e2 x − 18 = 0 Logarithmic Equations A. Definition A logarithmic equation is an equation containing a variable in a logarithmic expression. B. Using the Definition of a Logarithm to Solve Logarithmic Equations 1. Express the equation in the form log b M = c . 2. Use the definition of a logarithm to rewrite the equation in exponential form: log b M = c ⇔ b c = M . 3. Solve for the variable. 4. Check proposed solutions in the original equation. Include in the solution set only values for which C. Examples Solve for x. 1. log 5 x = 3 M >0. 2. log 6 (x + 5) + log 6 x = 2 3. log 2 (x − 3) + log 2 x − log 2 (x + 2) = 2 Solve, leaving your answer in terms of e. 4. ln x = 3 5. 6(ln(2x) = 30