Logarithmic Equations A logarithmic equation is an equation containing a variable in a logarithmic expression. Examples of logarithmic equations include Some logarithmic equations can be expressed in the form We can solve such equations by rewriting them in exponential form. log b M = c . log 4 1 x + 3 2 = 2 and ln 1 x + 2 2 - ln 1 4 x + 3 2 = ln a 1 x b . e 2 x - 8 e x + 7 = 0. 5 0, ln 3 6 . ln e x = x x = ln 3 ln e x = ln 3 x = 0 e x . e x = 3 e x = 1 e x - 3 = 0 or e x - 1 = 0 u = e x , u 2 - 4 u + 3 = 1 u - 3 21 u - 1 2 . 1 e x - 3 21 e x - 1 2 = 0 e 2 x - 4 e x + 3 = 0 u 2 - 4 u + 3 = 0. u = e x , e 2 x - 4 e x + 3 = 0. EXAMPLE 5 Use the definition of a logarithm to solve logarithmic equations. Using the Definition of a Logarithm to Solve Logarithmic Equations 1. Express the equation in the form 2. Use the definition of a logarithm to rewrite the equation in exponential form: 3. Solve for the variable. 4. Check proposed solutions in the original equation. Include in the solution set only values for which M 7 0. log b M=c means b c =M. Logarithms are exponents. log b M = c . Solving Logarithmic Equations Solve: a. b. Solution The form involves a single logarithm whose coefficient is 1 on one side and a constant on the other side. Equation (a) is already in this form. We will need to divide both sides of equation (b) by 3 to obtain this form. log b M = c 3 ln 1 2 x 2 = 12. log 4 1 x + 3 2 = 2 EXAMPLE 6
a. This is the given equation. Rewrite in exponential form: means Square 4. Subtract 3 from both sides. Check 13 : This is the given logarithmic equation. Substitute 13 for true because This true statement indicates that the solution set is b. This is the given equation. Divide both sides by 3. Rewrite the natural logarithm showing base This step is optional. Rewrite in exponential form: means Divide both sides by 2. Check This is the given logarithmic equation. Substitute for Simplify: Because ln we conclude ln true This true statement indicates that the solution set is Check Point 6 Solve: a. b. Logarithmic expressions are defined only for logarithms of positive real numbers. Always check proposed solutions of a logarithmic equation in the original equation. Exclude from the solution set any proposed solution that produces the logarithm of a negative number or the logarithm of 0. To rewrite the logarithmic equation in the equivalent exponential form we need a single logarithm whose coefficient is one. It is sometimes necessary to use properties of logarithms to condense logarithms into a single logarithm. In the next example, we use the product rule for logarithms to obtain a single logarithmic expression on the left side. Solving a Logarithmic Equation Solve: Solution This is the given equation. Use the product rule to obtain a single logarithm: log b M + log b N = log b 1 MN 2 . log 2 3 x 1 x - 7 24 = 3 log 2 x + log 2 1 x - 7 2 = 3 log 2 x + log 2 1 x - 7 2 = 3. EXAMPLE 7 b c = M , log b M = c 4 ln 1 3 x 2 = 8. log 2 1 x - 4 2 = 3 b e 4 2 r .