Exponential and Logarithmic Equations Study Guide

Concept Summary: Exponential and Logarithmic Equations

The study of exponential and logarithmic equations involves two primary principles defined by the Property of Equality. These properties provide the algebraic foundation for solving equations where variables are contained within exponents or within the arguments of logarithmic functions. The properties establish a two-way relationship, often denoted as "if and only if," meaning the implication works in both directions between the equality of the powers or logarithms and the equality of their internal components (exponents or arguments).

Property of Equality for Exponential Equations

The fundamental rule for equating exponential expressions states that if $b$ is a positive number other than $1$, then the expression $b^x = b^y$ is true if and only if $x = y$. This property can be understood through two specific linguistic interpretations. First, if two powers of the same base are equal, then their exponents must also be equal. Second, the converse is also true: if two exponents are equal, then the powers created using the same base are equal.

Numerical examples clarify this logic: If $2^x = 2^4$, then it follows that $x = 4$. Conversely, if $x = 4$, then it follows that $2^x = 2^4$.

Property of Equality for Logarithmic Equations

For logarithmic expressions, the property states that if $b$ is a positive number other than $1$, then $\log_b(x) = \log_b(y)$ if and only if $x = y$. This rule governs how logarithms with identical bases interact. In verbal terms, if two logarithms of the same base are equal to one another, then their arguments must be equal. Furthermore, if two arguments are equal and the bases applied to them are the same, then the resulting logarithms are equal.

Consider the following numerical demonstrations: If $\log_3(x) = \log_3(8)$, then it follows that $x = 8$. If $x = 8$, then it follows that $\log_3(x) = \log_3(8)$.

Do You UNDERSTAND? Conceptual Questions

1. ESSENTIAL QUESTION: How do properties of exponents and logarithms help you solve equations?

2. Vocabulary: Jordan claims that $x^2 + 3 = 12$ is an exponential equation. Is Jordan correct? Explain your thinking.

3. Communicate Precisely: How can properties of logarithms help to solve an equation such as $\log_6(8x - 2)^3 = 12$?

Do You KNOW HOW? Practice Problems

When solving the following equations, follow the instruction to round to the nearest hundredth if necessary and list any extraneous solutions that may arise during the algebraic process.

4. Solve the exponential equation: $16^{3x} = 256^{x+1}$

5. Solve the exponential equation: $6^{x+2} = 4^x$

6. Solve the logarithmic equation: $\log_5(x^2 - 44) = \log_5(7x)$

7. Solve the logarithmic equation: $\log_2(3x - 2) = 4$

8. Solve the exponential equation: $4^{2x} = 9^{x-1}$

9. Real-World Application: A rabbit farm had $200$ rabbits in the year $2015$. The population of rabbits is observed to increase by $30\%$ every year. Determine how many rabbits are on the farm in the year $2031$.