Textbook Question
Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 4^2x−1=64
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Ch. 4 - Exponential and Logarithmic Functions
Chapter 5, Problem 7
In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log7 (7/x)
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Properties of Logarithms
The properties of logarithms are rules that simplify the manipulation of logarithmic expressions. Key properties include the product rule (log_b(mn) = log_b(m) + log_b(n)), the quotient rule (log_b(m/n) = log_b(m) - log_b(n)), and the power rule (log_b(m^k) = k * log_b(m)). Understanding these properties is essential for expanding and simplifying logarithmic expressions.
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Change of Base Property
Quotient Rule of Logarithms
The quotient rule states that the logarithm of a quotient is the difference of the logarithms of the numerator and denominator. Specifically, log_b(m/n) = log_b(m) - log_b(n). This rule is particularly useful for breaking down complex logarithmic expressions into simpler components, making it easier to evaluate or expand them.
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3:49Product, Quotient, and Power Rules of Logs
Change of Base Formula
The change of base formula allows you to convert logarithms from one base to another, expressed as log_b(a) = log_k(a) / log_k(b) for any positive k. This is useful when dealing with logarithmic expressions that may not be easily evaluated in their original base, providing flexibility in calculations and simplifications.
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Related Practice
Textbook Question
In Exercises 1–10, approximate each number using a calculator. Round your answer to three decimal places. e^2.3
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Textbook Question
In Exercises 1–8, write each equation in its equivalent exponential form. log6 216 = y
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Textbook Question
In Exercises 5–9, graph f and g in the same rectangular coordinate system. Use transformations of the graph of f to obtain the graph of g. Graph and give equations of all asymptotes. Use the graphs to determine each function's domain and range. f(x) = e^x and g(x) = 2e^(x/2)
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Textbook Question
Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 32^x=8
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Textbook Question
In Exercises 1–10, approximate each number using a calculator. Round your answer to three decimal places. e^-0.95
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