Solve equations with variable exponents

Learning Objectives Identify an equation with a variable in the exponent. Rewrite numbers with a common base to prepare an equation for solving. Apply the one-to-one property of exponential functions (the 'same base' rule) to set exponents equal. Solve the resulting linear equation for the variable. Use the power of a power rule and the negative exponent rule to simplify expressions. Verify their solution by substituting it back into the original equation. If a single bacterium doubles every hour, how many hours would it take to have 4,096 bacteria? 🤔 Let's find out how to solve for time when it's the exponent! This tutorial will teach you a powerful algebraic technique for solving equations where the variable you're looking for is in the exponent....

TermDefinitionExample Exponential EquationAn equation in which a variable appears in the exponent.In the equation 5^x = 125, 'x' is a variable exponent, making this an exponential equation. BaseThe number that is being raised to a power in an exponential expression.In 2^8, the base is 2. Common BaseA number that can be used as the base for two or more different numbers in an equation by raising it to different powers.For the numbers 9 and 27, a common base is 3, because 9 = 3^2 and 27 = 3^3. One-to-One PropertyA property stating that if two exponential expressions with the same base are equal, then their exponents must also be equal.If 7^a = 7^b, then it must be true that a = b. Variable ExponentAn exponent that is a variable or contains a variable expression.In 3^(2x-1), the ex...

The Same Base Rule (One-to-One Property) If b^x = b^y, then x = y (where b > 0 and b ≠ 1) Use this rule after you have rewritten both sides of the equation to have the same base. It allows you to eliminate the bases and set the exponents equal to each other. Power of a Power Rule (b^m)^n = b^(m*n) Use this rule when a base raised to a power is itself raised to another power. You multiply the exponents together. Negative Exponent Rule b^(-n) = 1 / b^n Use this rule to handle fractions. It allows you to rewrite a fraction like 1/49 as an expression with a negative exponent, such as 7^(-2).