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Chapter 4: Problem 120

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

Short Answer

Expert verified
The relationship between a logarithmic equation (\( \log_b(x) = y \)) and an equivalent exponential equation (\( b^y = x \)) is that they are different representations of the same mathematical relationship. The output of a logarithmic function can serve as the exponent in the equivalent exponential function, and the exponent in an exponential equation can serve as the output in the equivalent logarithmic function.

Step by step solution

01

Understand Both Forms

Firstly, it is important to grasp the understanding of both forms. A logarithmic equation is typically written as \( \log_b(x) = y \), which can be read as 'log base b of x equals y'. An equivalent equation in exponential form is written as \( b^y = x \), read as 'b to the power of y equals x'.
02

Conversion to Exponential Form

To convert a logarithmic equation to an exponential form, take the base (b) of the logarithm, raise it to the power equal to the right side of the equation, and set it equal to the number inside the logarithm (x). For example, \( \log_b(x) = y \) becomes \( b^y = x \). This demonstrates that the output of a logarithmic function can serve as the exponent in its corresponding exponential function.
03

Conversion to Logarithmic Form

Conversely, you can convert an exponential equation to logarithmic form by taking the base (b) of the exponent, writing a logarithm with this base that equals the exponent (y), and the number the base is raised to (x) is placed inside the logarithm. For example, \( b^y = x \) becomes \( \log_b(x) = y \). This demonstrates that the exponent in an exponential equation can serve as the output in its corresponding logarithmic function.

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Most popular questions from this chapter

Check each proposed solution by direct substitution or with a graphing utility. $$(\log x)(2 \log x+1)=6$$

Make Sense? In Exercises \(73-76\), determine whether each statement makes sense or does not make sense, and explain your reasoning. When I used an exponential function to model Russia's declining population, the growth rate \(k\) was negative.

Without using a calculator, find the exact value of $$\frac{\log _{3} 81-\log _{\pi} 1}{\log _{2 \sqrt{2}} 8-\log 0.001}$$

In Exercises \(53-56,\) rewrite the equation in terms of base \(e\). Express the answer in terms of a natural logarithm and then round to three decimal places. $$ y=100(4.6)^{x} $$

Logarithmic models are well suited to phenomena in which growth is initially rapid but then begins to level off. Describe something that is changing over time that can be modeled using a logarithmic function.
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