In Exercises 93–102, solve each equation. 5^2x ⋅ 5^4x=125

Key Concepts

Exponential Equations

Exponential equations involve variables in the exponent and can often be solved by applying properties of exponents. In this case, the equation 5^(2x) ⋅ 5^(4x) can be simplified using the property that states a^m ⋅ a^n = a^(m+n). This allows us to combine the exponents before solving for the variable.

Properties of Exponents

The properties of exponents are rules that govern how to manipulate expressions involving powers. Key properties include the product of powers, which states that when multiplying like bases, you add the exponents, and the power of a power, which states that when raising a power to another power, you multiply the exponents. Understanding these properties is essential for simplifying and solving exponential equations.

Logarithms

Logarithms are the inverse operations of exponentiation and are used to solve equations where the variable is an exponent. For example, if you have an equation in the form a^x = b, you can use logarithms to isolate x by rewriting it as x = log_a(b). In this problem, once the exponential equation is simplified, logarithms may be used to find the value of x if necessary.