Exponential Equations

An exponential equation is an equation in which a variable appears in the exponent, typically in the form of a^x = b, where a is a positive constant, b is a positive number, and x is the variable.

The base in an exponential equation is the constant 'a' in the expression a^x, which is raised to the power of the variable x.

Convert 25 to the same base: 25 = 5^2. Then, set the exponents equal: -3x - 1 = 2. Solve for x: x = -1.

Convert 64 to the same base: 64 = 4^3. Set the exponents equal: p + 2 = 3. Solve for p: p = 1.

Convert 4 to base 2: 4 = 2^2. Then, 4^(x+1) = (2^2)^(x+1) = 2^(2x + 2). Set the exponents equal: x = 2x + 2. Solve for x: x = -2.

Convert 1/125 to the same base: 1/125 = 5^(-3). Set the exponents equal: a + 2 = -3. Solve for a: a = -5.

The property states that if a^m = a^n, then m = n, provided that a is not zero.

A logarithm is the inverse operation to exponentiation, answering the question: to what exponent must a base be raised, to produce a given number? For example, if a^b = c, then log_a(c) = b.

Convert 1/27 to the same base: 1/27 = 3^(-3). Set the exponents equal: x = -3.

64 can be expressed as 4^3, since 4 * 4 * 4 = 64.