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Chapter 4: Problem 6

Write each equation in its equivalent exponential form. $$3=\log _{b} 27$$

Short Answer

Expert verified
The equivalent exponential form of the equation \(\log_{b} 27 = 3\) is \(b^3 = 27\).

Step by step solution

01

Identify the components

Situate the parts of the log equation: \(\log _{b} 27=3\). Here, \(b\) is the base, 27 is the argument of the logarithm, and 3 is the value of the logarithm. Find the equivalent exponential form using the rule \(b^y=x\), where \(y\) is the logarithm value, \(b\) is the base, and \(x\) is the output when \(b\) is raised to \(y\).
02

Rewriting in exponential form

Input the identified components into the formula to create the equivalent exponential equation: \(b^3 = 27\). As such, the integer 3 becomes the exponent of \(b\), and 27 becomes the result. This exponential equation is equivalent to the given logarithmic equation \(\log _{b} 27 = 3\).

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Most popular questions from this chapter

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. $$ \log _{8}(13 \cdot 7) $$

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \log x+\log 15+\log \left(x^{2}-4\right)-\log (x+2) $$

Describe the power rule for logarithms and give an example.

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \log 5+\log 2 $$

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. $$ \ln \sqrt[2]{x} $$
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