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Chapter 9: Problem 51

Solve. Where appropriate, include approximations to three decimal places. If no solution exists, state this. $$ \log (2 x+1)=1 $$

Short Answer

Expert verified
x = 4.5

Step by step solution

01

Convert logarithmic equation to exponential form

Recall that the logarithmic equation \( \log_b(y) = x \) can be rewritten as \( y = b^x \). Here, we have \( \log (2x + 1) = 1 \. \) Since the base of the common logarithm is 10, we rewrite it as: \( \2x + 1 = 10^1 \).
02

Simplify the exponential equation

Simplify the right side of the equation: \( \2 x + 1 = 10 \).
03

Solve for x

Isolate \(x\) by subtracting 1 from both sides: \( \2 x = 9 \). Then, divide both sides by 2 to solve for \( x \: \) \( x = \frac{9}{2} = 4.5 \).
04

Verify the solution

Substitute 4.5 back into the original equation to ensure the solution is correct: \(\log (2(4.5) + 1) = \log (10) = 1 \), which is correct.

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Key Concepts

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

exponential_form
Understanding the exponential form of a logarithmic equation is crucial when solving logarithmic problems. A logarithmic equation such as \( \log_b(y) = x \) can be rewritten in exponential form as \( y = b^x \). This conversion helps simplify the equation and solve for the unknown variable. In our exercise, we have \( \log (2x + 1) = 1 \). Since the common logarithm (log) has a base of 10, this equation can be rewritten in exponential form as \( 2x + 1 = 10^1 \). Now, the equation is in a simpler form, allowing us to solve for \( x \) more easily.
common_logarithm
A common logarithm is a logarithm with a base of 10, indicated by \( \log \) without a base number. Understanding common logarithms is important when solving equations that involve \( \log \). In the given exercise, \( \log (2x + 1) = 1 \) is a common logarithm. This means we use a base of 10 to solve the equation. When we convert the logarithm to its exponential form, we get \( 2x + 1 = 10^1 \). This simplifies the process of solving for the unknown variable. Hence, one should be comfortable working with common logarithms to efficiently tackle such problems.
solving_equations
Solving logarithmic equations involves several steps, including converting the logarithm to exponential form and isolating the variable. Let's walk through the steps for our example, \( \log (2x + 1) = 1 \):
  • First, convert the logarithm to exponential form: \( 2x + 1 = 10^1 \).

  • Next, simplify the exponential equation: \( 10^1 \) simplifies to 10, giving us \( 2x + 1 = 10 \).

  • Then, solve for \( x \) by isolating it: subtract 1 from both sides to get \( 2x = 9 \).

  • Finally, divide by 2: \( x = \frac{9}{2} = 4.5 \).

  • Verify the solution by substituting \( x \) back into the original equation: \( \log (2(4.5) + 1) = \log (10) = 1 \), which confirms the solution is correct.

This step-by-step approach ensures you accurately solve logarithmic equations.

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Most popular questions from this chapter

The SenderBase "Security Network ranks e-mail volume using a logarithmic scale. The magnitude \(M\) of a network's daily e-mail volume is given by $$ M=\log \frac{v}{1.34} $$ where \(v\) is the number of e-mail messages sent each day. How many e-mail messages are sent each day by a network that has a magnitude of \(7.5 ?\) Data: forum.spamcop.net

Solve. If no solution exists, state this. $$ 1000^{2 x+1}=100^{3 x} $$

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Simplify. $$ \log _{1 / 4} \frac{1}{64} $$

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