key term - Power Property

Definition

The power property is a fundamental rule in exponents and logarithms that allows for the simplification and manipulation of expressions involving powers. It states that when raising a power to another power, the exponents can be multiplied together.

5 Must Know Facts For Your Next Test

1. The power property states that $(a^b)^c = a^{bc}$, where $a$, $b$, and $c$ are real numbers.
2. This property allows for the simplification of expressions involving multiple layers of exponents, making calculations more efficient.
3. The power property is widely used in solving exponential and logarithmic equations, as it helps to isolate the variable and transform the equation into a simpler form.
4. Understanding the power property is crucial for manipulating and simplifying expressions that contain exponents and logarithms.
5. The power property is also applicable to logarithms, where $ ext{log}_a(b^c) = c ext{log}_a(b)$.

Review Questions

• Explain how the power property can be used to simplify an expression involving exponents.
  • The power property states that $(a^b)^c = a^{bc}$, where $a$, $b$, and $c$ are real numbers. This means that when you have a power raised to another power, you can multiply the exponents together. For example, $(2^3)^4 = 2^{3 \times 4} = 2^{12}$. This allows you to simplify complex expressions with multiple layers of exponents by applying the power property to reduce the number of operations and make the expression easier to evaluate.
• Describe how the power property can be used to solve exponential equations.
  • When solving exponential equations, the power property can be used to isolate the variable and transform the equation into a simpler form. For example, to solve the equation $2^{3x} = 32$, you can apply the power property to rewrite the equation as $2^{3x} = 2^5$. Since the bases are the same, you can set the exponents equal to each other: $3x = 5$, which can then be solved to find the value of $x$. The power property is crucial for manipulating exponential equations and simplifying them to find the unknown variable.
• Explain how the power property can be applied to logarithmic expressions.
  • The power property also applies to logarithms, where $ ext{log}_a(b^c) = c ext{log}_a(b)$. This means that the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the base. For example, $ ext{log}_2(8^3) = 3 ext{log}_2(8)$. This property is useful for simplifying and evaluating logarithmic expressions, as well as for solving logarithmic equations by isolating the variable using the power property.

Related terms

Exponent:

A number that represents the power to which a base number is raised.

Logarithm:

The inverse operation of exponents, used to determine the power to which a base number must be raised to get a certain value.

Base:

The number that is being raised to a power or whose logarithm is being taken.