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Chapter 4: Problem 22

Write the equation in exponential form. $$ \log \left(\frac{1}{1,000,000}\right)=-6 $$

Short Answer

Expert verified
The exponential form of \(\log \left(\frac{1}{1,000,000}\right) = -6\) is \(10^{-6} = \frac{1}{1,000,000}\).

Step by step solution

01

Understand the Logarithmic Form

The given logarithmic equation is \(\log \left(\frac{1}{1,000,000}\right) = -6\). This is in the form \(\log_b(x) = y\).
02

Express in Terms of Base 10

Since no base is specified, it's understood to be base 10. So the equation becomes \(\log_{10} \left(\frac{1}{1,000,000}\right) = -6\).
03

Convert to Exponential Form

To convert from logarithmic to exponential form, use the relationship \(b^y = x\). Here, \(b = 10\), \(y = -6\), and \(x = \frac{1}{1,000,000}\).
04

Write Exponential Equation

Based on the relationship from Step 3, the exponential form of the equation is \(10^{-6} = \frac{1}{1,000,000}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

logarithmic equations
Logarithmic equations are mathematical statements that involve logarithms. Logarithms are the inverse operations to exponents.
They help associate a number with the power it must be raised to obtain another number. For example, in the equation \(\text{log}_b(x) = y\), \(\text{log}\) stands for logarithm, \(b\) is the base, and \(x\) is the number we want to find the log of, while \(y\) is the exponent to which the base must be raised to produce \(x\).
By solving logarithmic equations, we translate the logarithmic terms back into their exponential forms. This is very useful in various fields, including science and engineering.
base 10
When dealing with logarithms, the base is an important concept. The base of a logarithm tells us the number that is raised to a power.
A common base used in many applications is base 10, which is often implied and does not need to be explicitly written as \(\text{log}_{10}\). It is also referred to as the common logarithm.
  • Base 10 logarithms are common in calculations involving scientific data and are also used to measure the intensity of earthquakes (Richter scale) and sound (decibels).
Understanding that base 10 is implied when no base is specified makes problems involving logarithms easier to solve.
exponents
Exponents indicate how many times a number, known as the base, is multiplied by itself. For instance, \(10^2 = 10 \times 10 = 100\).
In this example, 10 is the base, and 2 is the exponent or power. Exponents are shortcuts for repeated multiplication.
  • When writing an equation such as \(10^{-6}\), it represents taking the reciprocal of the base raised to the positive exponent: \(10^{-6} = 1/10^6 = 1/1,000,000\).
Converting logarithmic equations into their exponential form helps clearly see how the numbers relate. This conversion uses the fundamental relationship: \(b^y = x\).

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Most popular questions from this chapter

Use the model \(A=P e^{r t} .\) The variable \(A\) represents the future value of \(P\) dollars invested at an interest rate \(r\) compounded continuously for \(t\) years. If a couple has \(\$ 80,000\) in a retirement account, how long will it take the money to grow to \(\$ 1,000,000\) if it grows by \(6 \%\) compounded continuously? Round to the nearest year.

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(1024=19^{x}+4\)

Determine if the statement is true or false. For each false statement, provide a counterexample. For example, \(\log (x+y) \neq \log x+\log y\) because \(\log (2+8) \neq \log 2+\log 8\) (the left side is 1 and the right side is approximately 1.204 ). $$ \log _{5}\left(\frac{1}{x}\right)=\frac{1}{\log _{5} x} $$

A table of data is given. a. Graph the points and from visual inspection, select the model that would best fit the data. Choose from $$\begin{array}{ll} y=m x+b \text { (linear) } & y=a b^{x} \text { (exponential) } \\ y=a+b \ln x \text { (logarithmic) } & y=\frac{c}{1+a e^{-b x}} \text { (logistic) } \end{array}$$ b. Use a graphing utility to find a function that fits the data. $$ \begin{array}{|c|c|} \hline x & y \\ \hline 3 & 2.7 \\ \hline 7 & 12.2 \\ \hline 13 & 25.7 \\ \hline 15 & 30 \\ \hline 17 & 34 \\ \hline 21 & 44.4 \\ \hline \end{array} $$

Prove the power property of logarithms: \(\log _{b} x^{p}=p \log _{b} x\)
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