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Chapter 4: Problem 88

Describe the relationship between an equation in logarithmic form and an equivalent equation in exponential form.

Short Answer

Expert verified
A logarithmic equation \( \log_b(a) = c \) and an exponential equation \( b^c = a \) represent the same relationship between \( a \), \( b \) and \( c \). The two forms are simply different ways of expressing the same mathematical truth. The base and result in a logarithm become the base and outcome in an exponential equation, respectively, with the exponent (in exponential form) being the result in the logarithmic form.

Step by step solution

01

Understanding Logarithmic and Exponential Forms

First, it's important to grasp the basic forms of these two types of equations. A logarithmic equation is generally given in the form: \( \log_b(a) = c \), where \( b \) is the base, \( a \) is the argument, and \( c \) is the result. On the other hand, an exponential equation is commonly given in the form: \( b^c = a \), where \( b \) is the base, \( c \) is the exponent, and \( a \) is the result.
02

Converting Logarithmic Form to Exponential Form

To convert a logarithmic equation to an exponential equation, you can understand \( \log_b(a) = c \) to mean that 'the base \( b \) raised to the power \( c \) equals \( a \)' which gives us the exponential form \( b^c = a \). This interpretation comes from the definition of a logarithm.
03

Converting Exponential Form to Logarithmic Form

Similarly, to translate an exponential equation to a logarithmic one, the equation \( b^c = a \) is seen as '\( c \) is the power to which the base \( b \) must be raised to get \( a \)' which provides us with the logarithmic form \( \log_b(a) = c \). Again, this interpretation arises from the definition of an exponential function.

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Most popular questions from this chapter

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. $$ \log _{3} 405-\log _{3} 5 $$

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. $$ 2 \ln x-\frac{1}{2} \ln y $$

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. $$ \log M^{-8} $$

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. $$ \log _{b}\left(x y^{3}\right) $$

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. $$ \ln x+\ln 7 $$
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