key term - Like Bases

Definition

Like bases refer to exponential expressions that have the same base. This is an important concept in the context of solving exponential and logarithmic equations, as the properties of exponents and logarithms can be leveraged to simplify and solve these types of equations.

5 Must Know Facts For Your Next Test

1. When dealing with exponential equations with like bases, the exponents can be combined using the power rule, which states that $a^m \cdot a^n = a^{m+n}$.
2. Logarithms can be used to solve exponential equations with like bases by applying the power rule in reverse, converting the exponential equation to a linear equation in the logarithmic form.
3. Exponential equations with like bases can also be solved by setting the exponents equal to each other and solving for the unknown variable.
4. Simplifying expressions with like bases involves combining the exponents using the power rule, which can help in solving more complex exponential and logarithmic equations.
5. Understanding the properties of like bases is crucial in manipulating and solving a variety of exponential and logarithmic equations encountered in college algebra.

Review Questions

• Explain how the power rule for exponents can be used to simplify expressions with like bases.
  • The power rule for exponents states that $a^m \cdot a^n = a^{m+n}$. This means that when dealing with exponential expressions that have the same base, the exponents can be combined by adding them together. This simplification process is particularly useful when solving exponential equations with like bases, as it allows you to isolate the unknown variable and solve for its value.
• Describe how logarithms can be used to solve exponential equations with like bases.
  • Exponential equations with like bases can be converted to linear equations by applying logarithms to both sides of the equation. This is because the logarithm of an exponential expression is equal to the exponent. For example, if we have the equation $a^x = b$, where $a$ and $b$ are constants, we can take the logarithm of both sides to get $ ext{log}_a(a^x) = ext{log}_a(b)$, which simplifies to $x = ext{log}_a(b)$. This transformation allows us to solve for the unknown variable $x$ in the original exponential equation.
• Analyze the importance of understanding the properties of like bases in the context of solving exponential and logarithmic equations.
  • Understanding the properties of like bases is crucial in the context of solving exponential and logarithmic equations because it allows you to manipulate and simplify these types of equations. By recognizing that expressions with the same base can be combined using the power rule, you can isolate the unknown variable and solve for its value. Additionally, the relationship between exponents and logarithms, where logarithms can be used to convert exponential equations into linear equations, relies on the concept of like bases. Mastering the properties of like bases equips you with the necessary tools to tackle a wide range of exponential and logarithmic equations encountered in college algebra.

Related terms

Exponents:

A mathematical notation that represents repeated multiplication of the same number. Exponents are used to express powers, such as $a^n$, where $a$ is the base and $n$ is the exponent.

Logarithms:

The inverse operation of exponents, used to express the power to which a base must be raised to get a certain number. Logarithms are written as $ ext{log}_a(x)$, where $a$ is the base and $x$ is the value.

Exponential Equations:

Equations that contain one or more exponential expressions, such as $a^x = b$, where $a$ and $b$ are constants, and $x$ is the unknown variable.