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Write each equation in its equivalent logarithmic form. 2-4 = 1/16
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Ch. 4 - Exponential and Logarithmic Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 5, Problema 13
In Exercises 13–15, write each equation in its equivalent exponential form. 1/2 = log49 7
Guida verificata passo dopo passo1
Understand that the equation \( \frac{1}{2} = \log_{49} 7 \) is in logarithmic form, where \( \log_b a = c \) means \( b^c = a \).
Identify the base \( b \), the result \( a \), and the exponent \( c \) from the logarithmic equation. Here, \( b = 49 \), \( a = 7 \), and \( c = \frac{1}{2} \).
Rewrite the equation in exponential form using the identified components: \( 49^{\frac{1}{2}} = 7 \).
Recognize that rewriting in exponential form involves expressing the base raised to the power of the exponent equals the result.
Verify the conversion by checking if the exponential form correctly represents the original logarithmic equation.
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Logarithmic Functions
Logarithmic functions are the inverse of exponential functions. They express the power to which a base must be raised to obtain a certain number. For example, in the equation log_b(a) = c, b is the base, a is the result, and c is the exponent. Understanding this relationship is crucial for converting between logarithmic and exponential forms.
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5:26
Graphs of Logarithmic Functions
Exponential Form
Exponential form refers to expressing a number as a base raised to a power. For instance, the equation a = b^c indicates that b is raised to the power of c to yield a. In the context of logarithms, converting a logarithmic equation to exponential form allows us to solve for unknown variables by rewriting the equation in a more straightforward manner.
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6:13Exponential Functions
Change of Base Formula
The change of base formula is a method used to convert logarithms from one base to another, which can simplify calculations. It states that log_b(a) can be expressed as log_k(a) / log_k(b) for any positive k. This concept is particularly useful when dealing with logarithms that do not have a readily available base, allowing for easier computation and understanding of logarithmic relationships.
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5:36Change of Base Property
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