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 exponential equation 5^{-3x} = 125?

Back

To solve 5^{-3x} = 125, rewrite 125 as 5^3. This gives you 5^{-3x} = 5^3. Since the bases are the same, set the exponents equal: -3x = 3, leading to x = -1.

Front

What is the first step in solving the equation 4^(p+2) = 64?

Back

The first step is to express 64 as a power of 4. Since 64 = 4^3, you can rewrite the equation as 4^(p+2) = 4^3 and then set the exponents equal.

Front

How do you convert the equation 10^(2x+4) = 100000 into a simpler form?

Back

Rewrite 100000 as a power of 10. Since 100000 = 10^5, the equation becomes 10^(2x+4) = 10^5, allowing you to set the exponents equal.

Front

What is the solution to the equation 5^{-3x - 1} = 25?

Back

Rewrite 25 as 5^2. The equation becomes 5^{-3x - 1} = 5^2. Set the exponents equal: -3x - 1 = 2, leading to x = -1.

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

How do you solve for x in the equation 8 = 2^{5x+7}?

Back

Rewrite 8 as a power of 2: 8 = 2^3. The equation becomes 2^3 = 2^{5x+7}. Set the exponents equal: 3 = 5x + 7, leading to x = -4/5.

Front

What is the property of exponents that allows you to set exponents equal when the bases are the same?

Back

This property is known as the equality of exponents property, which states that if a^m = a^n (where a > 0 and a ≠ 1), then m = n.

Front

What is the logarithmic form of the exponential equation 2^x = 16?

Back

The logarithmic form is x = log_2(16). Since 16 = 2^4, this simplifies to x = 4.

Front

How can you check your solution for an exponential equation?

Back

You can check your solution by substituting the value back into the original equation to see if both sides are equal.