Exponential and Logarithmic Equations and Inequalities
Tom Green
8 min read
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Study Guide Overview
This study guide covers exponential and logarithmic equations and inequalities. It explains how to use properties of exponents and logarithms, including the product, quotient, and power rules, and the inverse relationship between these functions to solve equations and inequalities. It also emphasizes checking for extraneous solutions. Finally, it covers finding the inverse of exponential and logarithmic functions, including transformations like horizontal and vertical shifts and stretches/compressions.
#2.13 Exponential and Logarithmic Equations and Inequalities
#Approaching Equations and Inequalities of Exponents & Logs
Properties of exponents, properties of logarithms, and the inverse relationship between exponential and logarithmic functions can be used to simplify and solve equations and inequalities involving exponents and logarithms. These properties can be used to change the form of the equation or inequality, making it easier to solve. 💕
#🪄 The Magic Behind Properties!
Some examples of properties of exponents that can be used to simplify equations include the product of powers property, the quotient of powers property, and the power of a power property. For example, if we have the equation 2^x * 2^y = 2^z, we can use the product of powers property to simplify it to 2^(x+y) = 2^z. 🎩
Similarly, properties of logarithms such as the product rule, quotient rule, and power rule can be used to simplify equations and inequalities involving logarithms. For example, if we have the equation log_2(x) + log_2(y) = log_2(xy), we can use the product rule to simplify it to log_2(x) + log_2(y) = log_2(x) + log_2(y)
The inverse relationship between exponential and logarithmic functions can also be used to solve equations and inequalities. For example, if we have the equation 2^x = 8, we can use the inverse relationship to write it as log_2(8) = x, and then solve for x.
Key Log Properties
- Product Rule:
log_b(MN) = log_b(M) + log_b(N) - Quotient Rule:
log_b(M/N) = log_b(M) - log_b(N) - Power Rule:
log_b(M^p) = p * log_b(M) - Change of Base:
log_b(M) = log_c(M) / log_c(b)
Think of logs as 'undoing' exponents. If you see a log equation, try to isolate the log term and then rewrite it in exponential form to solve...
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