Textbook Question
In Exercises 1–8, write each equation in its equivalent exponential form. log6 216 = y
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Ch. 4 - Exponential and Logarithmic Functions
Chapter 5, Problem 9
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
Verified Solution
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Exponential Equations
Exponential equations are mathematical expressions where variables appear in the exponent. To solve these equations, one common method is to express both sides with the same base, allowing for the exponents to be equated. This approach simplifies the problem and makes it easier to isolate the variable.
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Base and Exponent
In an exponential expression, the base is the number that is raised to a power, while the exponent indicates how many times the base is multiplied by itself. Understanding the relationship between bases and exponents is crucial for manipulating exponential equations, as it allows for the conversion of different exponential forms into a common base.
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7:39Introduction to Exponent Rules
Equating Exponents
When both sides of an exponential equation are expressed with the same base, the next step is to equate the exponents. This principle arises from the property that if a^m = a^n (where a is the base), then m must equal n. This allows for straightforward solutions to the variable in the exponent.
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Related Practice
Textbook Question
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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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
In Exercises 1–10, approximate each number using a calculator. Round your answer to three decimal places. e^-0.95
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Textbook Question
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.
log(x/100)
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Textbook Question
Use the compound interest formulas to solve Exercises 10–11. Suppose that you have $5000 to invest. Which investment yields the greater return over 5 years: 1.5% compounded semiannually or 1.45% compounded monthly?
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