Solving Exponential Equations

Front

What is an exponential equation?

Back

An exponential equation is an equation in which a variable appears in the exponent. For example, in the equation 2^x = 8, x is the variable in the exponent.

Front

How do you solve the equation 16^(x+5) = 64^(-2)?

Back

Convert both sides to the same base: 16 = 2^4 and 64 = 2^6. Thus, 2^(4(x+5)) = 2^(6(-2)). This simplifies to 4(x+5) = -12, leading to x = -8.

Front

What is the first step in solving exponential equations?

Back

The first step is often to express both sides of the equation with the same base, if possible.

Front

How do you solve the equation 16^(2x) = 256^(2x+5)?

Back

Convert both sides to the same base: 16 = 2^4 and 256 = 2^8. This gives 2^(4(2x)) = 2^(8(2x+5)). Simplifying leads to 8x = 40, so x = -5.

Front

What is the solution to the equation 4^(p+2) = 64?

Back

Convert 64 to base 4: 64 = 4^3. Thus, p+2 = 3, leading to p = 1.

Front

What is the relationship between the bases in exponential equations?

Back

If two exponential expressions with the same base are equal, then their exponents must also be equal.

Front

How do you solve the equation 9^(8-x) = 27^(x-3)?

Back

Convert both sides to the same base: 9 = 3^2 and 27 = 3^3. This gives 3^(2(8-x)) = 3^(3(x-3)). Simplifying leads to 16 - 2x = 3x - 9, so x = 5.

Front

What is the base of an exponential function?

Back

The base of an exponential function is the constant that is raised to a power. In the function f(x) = a^x, 'a' is the base.

Front

What does it mean to isolate the variable in an exponential equation?

Back

Isolating the variable means rearranging the equation so that the variable is on one side and all other terms are on the other side.

Front

What is the solution to the equation 16^(x+5) = 64^(-2)?

Back

The solution is x = -8.