Like bases refer to exponential expressions that have the same base value. These expressions can be combined or simplified by applying the properties of exponents, allowing for more efficient manipulation and analysis of exponential and logarithmic equations.
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Like bases can be combined by adding the exponents when the bases are the same.
The power rule for like bases states that $b^m \cdot b^n = b^{m+n}$, where $b$ is the common base.
Exponential equations with like bases can be solved by setting the exponents equal to each other and solving for the unknown variable.
Logarithmic expressions with like bases can be simplified by subtracting the exponents, as $\log_b(x) - \log_b(y) = \log_b(x/y)$.
Understanding the properties of like bases is crucial for solving exponential and logarithmic equations efficiently.
Review Questions
Explain how the power rule for like bases can be used to simplify exponential expressions.
The power rule for like bases states that $b^m \cdot b^n = b^{m+n}$, where $b$ is the common base. This allows you to combine exponential expressions with the same base by adding the exponents. For example, $2^3 \cdot 2^5 = 2^{3+5} = 2^8$. This simplification is useful when working with exponential equations and functions, as it reduces the number of terms and makes the expressions easier to manipulate.
Describe how the properties of like bases can be used to solve exponential equations.
When solving exponential equations with like bases, you can set the exponents equal to each other and solve for the unknown variable. For example, to solve the equation $2^x = 2^5$, you can set the exponents equal: $x = 5$. This works because the bases are the same, so the only way for the expressions to be equal is if the exponents are equal. Understanding this property of like bases is crucial for efficiently solving a variety of exponential equations.
Analyze how the logarithm property of like bases can be used to simplify logarithmic expressions.
The logarithm property of like bases states that $\log_b(x) - \log_b(y) = \log_b(x/y)$. This means that logarithmic expressions with the same base can be simplified by subtracting the exponents. For example, $\log_3(81) - \log_3(9) = \log_3(81/9) = \log_3(9)$. This property is useful when working with logarithmic functions and equations, as it allows you to manipulate the expressions more easily and arrive at the desired solution.