Exponential & Logarithmic Equations Exponential Equations One-to-One Exponential Equations with Different Bases Logarithmic Equations One-to-One Other Logarithmic Equations

Exponential Equations (One-to-One) In an Exponential Equation, the variable is in the exponent. There may be one exponential term or more than one, like… If you can isolate terms so that the equation can be written as two expressions with the same base, as in the equations above, then the solution is simple. or

Exponential Equations (One-to-One)   1. Isolate the exponential expression and rewrite the constant in terms of the same base. 2. Set the exponents equal to each other (drop the bases) and solve the resulting equation.

Exponential Equations (One-to-One) Example #2 - Two exponential expressions. 3x+1 = 9x-2 3x+1 = (32)x-2 3x+1 = 32x-4 x + 1 = 2x – 4 x = 5 1. Isolate the exponential expressions on either side of the =. We then rewrite the 2nd expression in terms of the same base as the first. 2. Set the exponents equal to each other (drop the bases) and solve the resulting equation.

Exponential Equations with Different Bases The Exponential Equations below contain exponential expressions whose bases cannot be rewritten as the same rational number. The solutions are irrational numbers, we will need to use a log function to evaluate them. or

Exponential Equations with Different Bases   1. Isolate the exponential expression. 2. Rewrite the equation in log form. 3. Use your calculator and perhaps base change formula to solve.

Logarithmic Equations In a logarithmic equation, the variable can be inside the log function or inside the base of the log. There may be one log term or more than one. For example …

Logarithmic Equations (One-to-One) If there is exactly one log on each side, and the bases are the same, you can use the one-one property to say that the expressions are the same. However, you must check for extraneous solutions because you cannot take the log of a negative number.

Example 4 Solve this one-to-one equation. log6(x2 – 28) = log6(3x) x2 – 28 = 3x x2 - 3x – 28 = 0 (x-7)(x+4) = 0 x = 7 or x = -4 Only x = 7 works.

Logarithmic Equations Example 5 - Variable inside the log function. log4(2x – 1) + 3 = 5 log4 (2x – 1) = 2 42 = 2x – 1 16 = 2x – 1 2x = 17 x = 8.5 1. Isolate the log expression. 2. Rewrite the log equation as an exponential equation and solve for ‘x’.

Logarithmic Equations   1. Rewrite the log equation as an exponential equation. 2. Solve the exponential equation.

Logarithmic Equations Example 7 Combining logs log x + log (x + 6) = log 16 log {x(x+6)} = log 16 x(x + 6) = 16 x2 + 6x = 16 x2 + 6x – 16 = 0 (x + 8)(x – 2) = 0 x = -8 or x = 2, but only x = 2 works