Find the missing exponent or base

Learning Objectives Solve exponential equations where the variable is the exponent by creating a common base. Solve equations where the variable is the base by using roots and fractional exponents. Apply the properties of negative and fractional exponents to solve for a missing variable. Differentiate between problems requiring finding a missing base versus a missing exponent and apply the correct algebraic strategy. Simplify complex exponential expressions to isolate the term containing the unknown variable. Translate simple word problems involving exponential relationships into solvable equations. How do scientists determine the age of a fossil or a bank calculate how long it takes for an investment to double? 🤯 It all comes down to solving for a missing piece in an expon...

TermDefinitionExample BaseIn a power, the base is the number that is repeatedly multiplied.In `5^3`, the number 5 is the base. ExponentIn a power, the exponent (or index) indicates how many times to multiply the base by itself.In `5^3`, the number 3 is the exponent. Exponential EquationAn equation in which a variable appears in the position of an exponent.`3^x = 81` is an exponential equation. Common BaseA number that can be used as the base for two or more powers in an equation, allowing for comparison.To solve `4^x = 8`, we can use a common base of 2, since `4 = 2^2` and `8 = 2^3`. RootThe inverse operation of raising a number to a power. The nth root of a number 'a' is a number 'b' such that `b^n = a`.The cube root of 64 is 4, because `4^3 = 64`. Fractional Exponent...

Equality of Powers (Common Base Rule) If `b^x = b^y`, then `x = y` (where `b > 0` and `b ≠ 1`) Use this rule when the variable is in the exponent. The goal is to rewrite both sides of the equation with the same base. Once the bases are identical, you can set the exponents equal to each other and solve. Solving for the Base If `x^n = a`, then `x = a^(1/n)` or `x = \sqrt[n]{a}` Use this rule when the variable is in the base. To isolate the variable, you must take the nth root of both sides of the equation. Remember that if 'n' is an even integer, there may be both a positive and a negative solution. Negative Exponent Rule `b^(-n) = 1 / b^n` This rule is essential when one side of the equation is a fraction. It allows you to rewrite a fraction as a power wi...