/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 17 Solve for \(x\). a. \(\log x=3... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 17

Solve for \(x\). a. \(\log x=3\) d. \(\log (x+1)-\log x=1\) b. \(\log (x+1)=3\) e. \(\log x-\log (x+1)=1\) c. \(3 \log x=5\)

Short Answer

Expert verified
a. \( x = 1000 \), b. \( x = 999 \), c. \( x = 10^{1.6667} \), d. \( x = \frac{1}{9} \), e. No solution

Step by step solution

01

Step 1a: Solve for x in \( \log x = 3 \)

Rewrite the logarithmic equation in its exponential form: \( x = 10^3 \). So, \( x = 1000 \).
02

Step 1b: Solve for x in \( \log (x+1) = 3 \)

Rewrite the logarithmic equation in its exponential form: \( x + 1 = 10^3 \). So, \( x + 1 = 1000 \). Subtract 1 from both sides: \( x = 999 \).
03

Step 1c: Solve for x in \( 3 \log x = 5 \)

Divide by 3 on both sides: \( \log x = \frac{5}{3} \). Rewrite the logarithmic equation in its exponential form: \( x = 10^{5/3} \). Simplify: \( x = 10^{1.6667} \).
04

Step 1d: Solve for x in \( \log (x+1) - \log x = 1 \)

Use the property \( \log a - \log b = \log\left(\frac{a}{b}\right) \): \( \log \left(\frac{x+1}{x}\right) = 1 \). Rewrite the logarithmic equation in its exponential form: \( \frac{x+1}{x} = 10^1 \). Simplify: \( \frac{x+1}{x} = 10 \). Multiply both sides by \( x \: x + 1 = 10x \). Rearrange the equation: \( x + 1 = 10x \: 1 = 9x \). Solve for \( x \: x = \frac{1}{9} \).
05

Step 1e: Solve for x in \( \log x - \log (x+1) = 1 \)

Use the property \( \log a - \log b = \log\left(\frac{a}{b}\right) \): \( \log\left(\frac{x}{x+1}\right) = 1 \). Rewrite the logarithmic equation in its exponential form: \( \frac{x}{x+1} = 10^1 \). Simplify: \( \frac{x}{x+1} = 10 \). Multiply both sides by \( x + 1 \: x = 10(x+1) \). Expand and rearrange the equation: \( x = 10x + 10 \: -9x = 10 \). Solve for \( x \: x = -\frac{10}{9} \). Since the log of a negative number is undefined, there is no solution \( x \).

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Key Concepts

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

exponential form
Understanding exponential form is crucial when working with logarithmic equations. When you have an equation in the form of \(\text{log}_b(x) = y\), you can convert it into its exponential form which is \((b^y = x)\). This conversion is really useful for solving logarithmic equations.
For example, in the problem \(\text{log}(x) = 3\):
  • We can rewrite it in exponential form as \10^3 = x\.
  • Therefore, \x = 1000\.
By rewriting logarithmic equations in exponential form, you often simplify the problem, making it easier to isolate the variable you are solving for.
logarithmic properties
Logarithmic properties provide handy tools that simplify complex log expressions. Key properties include:
  • Product Property: \(\text{log}_b(xy) = \text{log}_b(x) + \text{log}_b(y)\)
  • Quotient Property: \(\text{log}_b\frac{x}{y} = \text{log}_b(x) - \text{log}_b(y)\)
  • Power Property: \(\text{log}_b(x^y) = y \text{log}_b(x)\)
Applying these properties helps to simplify and solve logarithmic equations.
For instance, in the equation \(\text{log}(x+1) - \text{log}(x) = 1\), we can use the Quotient Property:
  • Rewrite it as \(\text{log}\frac{x+1}{x} = 1\).
  • Next, convert it to exponential form: \(\frac{x+1}{x} = 10^1\).
  • Solving that leads to \x = \frac{1}{9}\.
Understanding and applying these properties is a useful strategy to tackle more complicated logarithmic problems.
solving log equations
Solving logarithmic equations often involves a few specific steps aimed at isolating the variable. Here's a general guide to solving log equations:
  • Step 1: Combine and simplify logarithms using logarithmic properties if necessary. For example, if you have \(\text{log}(x+1) - \text{log}(x+1) = 1\), use the Quotient Property to combine them into one log.
  • Step 2: Convert the simplified logarithmic expression to its exponential form. For example, \(\text{log}_b(x) = y\) can be rewritten as \b^y = x\.
  • Step 3: Solve for the variable usually by isolating it through basic algebraic operations. For example, if \(\frac{x}{x+1} = 10\), you would multiply both sides by \(x+1\) and solve for x.
Let's look at an example: \(3 \text{log}(x) = 5\).
  • We start by isolating the log term: \(\text{log}(x) = \frac{5}{3}\).
  • Then, convert to exponential form: \(x = 10^{5/3}\).
  • Simplify to find that \(x \approx 21.5443\).
By consistently applying these steps, solving logarithmic equations becomes much more manageable.

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Most popular questions from this chapter

Form the exponential function from its logarithmic equivalent for each of the following. a. \(\log y=\log 1400+(\log 1.06) x\) b. \(\log y=\log (25,000)+(\log 0.87) x\) c. \(\log y=2+(\log 2.5) x\) d. \(\log y=4.25+(\log 0.63) x\)

A city of population 1.5 million is expected to experience a \(15 \%\) decrease in population every 10 years. a. What is the 10 -year decay factor? What is the yearly decay factor? The yearly decay rate? b. Use part (a) to create an exponential population model \(g(t)\) that gives the population (in millions) after \(t\) years. c. Create an exponential population model \(h(t)\) that gives the population (in millions) after \(t\) years, assuming a \(1.625 \%\) continuous yearly decrease. d. Compare the populations predicted by the two functions after 20 years. What can you conclude?

Use the accompanying table to estimate the number of years it would take \(\$ 100\) to become \(\$ 300\) at the following interest rates compounded annually. a. \(3 \%\) b. \(7 \%\) Compound Interest over 40 Years $$ \begin{array}{ccc} \hline \text { Number of } & \text { Value of } \$ 100 \text { at } & \text { Value of } \$ 100 \text { at } \\ \text { Years } & 3 \%(\$) & 7 \%(\$) \\ \hline 0 & 100 & 100 \\ 10 & 134 & 197 \\ 20 & 181 & 387 \\ 30 & 243 & 761 \\ 40 & 326 & 1497 \\ \hline \end{array} $$

The effective annual interest rate on an account compounded continuously is \(3.38 \%\). Estimate the nominal interest rate.

Determine which data sets (if any) describe \(y\) as an exponential function of \(x\), then construct the exponential function. a. $$ \begin{aligned} &\text { a. }\\\ &\begin{array}{cc} \hline x & \log y(\text { or } Y) \\ \hline 0 & 2.30103 \\ 10 & 4.30103 \\ 20 & 4.90309 \\ 30 & 5.25527 \\ 40 & 5.50515 \\ \hline \end{array} \end{aligned} $$ b. $$ \begin{array}{rr} \hline x & \log y \text { (or } Y \text { ) } \\ \hline 0 & 4.77815 \\ 10 & 3.52876 \\ 20 & 2.27938 \\ 30 & 1.02999 \\ 40 & -0.21945 \\ \hline \end{array} $$
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