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Chapter 12: Problem 94

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 and an exponential equation is that they are inverse operations of each other. Given the logarithmic equation \(\log_b (y) = x\), the equivalent exponential form is \(b^x = y\). Conversely, given the exponential equation \(b^x = y\), the equivalent logarithmic form is \( \log_b (y) = x\).

Step by step solution

01

Definition of Logarithm

A logarithm is an exponent applied to a specific base. The logarithm of a number is the exponent to which another fixed number, the base, must be raised to produce that number. If \(y = b^x\) then the logarithm base \(b\) of \(y\) is \(x\), denoted as \( \log_b (y) = x\).
02

From log to exponential form

We can convert from logarithmic form to exponential form. Given a logarithm equation \( \log_b (y) = x\), we can say that \(b^x = y\). This is the equivalent exponential form of the logarithm equation.
03

From exponential to log form

Likewise, we can convert an exponential form equation to a logarithmic form. Given an exponential equation \(b^x = y\), we can rewrite this equation in logarithmic form as \( \log_b (y) = x\).
04

Example conversion

As an example, consider the exponential equation \(2^3 = 8\). Its equivalent logarithmic form will be \( \log_2 (8) = 3\). Conversely, if we have the logarithmic equation \( \log_5 (25) = 2\), its equivalent exponential form will be \(5^2 = 25\).

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Most popular questions from this chapter

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

a. Use a graphing utility (and the change-of-base property) to graph \(y=\log _{3} x\) b. Graph \(\quad y=2+\log _{3} x, \quad y=\log _{3}(x+2), \quad\) and \(y=-\log _{3} x \quad\) in the same viewing rectangle as \(y=\log _{3} x .\) Then describe the change or changes that need to be made to the graph of \(y=\log _{3} x\) to obtain each of these three graphs.

a. Simplify: \(e^{\ln 3}\) b. Use your simplification from part (a) to rewrite \(3^{x}\) in terms of base \(e\)

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The loudness level of a sound, \(D,\) in decibels, is given by the formula $$D=10 \log \left(10^{12} I\right)$$ where I is the intensity of the sound, in watts per meter \(^{2} .\) Decibel levels range from \(0,\) a barely audible sound, to \(160,\) a sound resulting in a nuptured eardrum. Use the formula to solve Exercises. The sound of a blue whale can be heard 500 miles away, reaching an intensity of \(6.3 \times 10^{6}\) watts per meter? Determine the decibel level of this sound. At close range, can the sound of a blue whale rupture the human eardrum?
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