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Chapter 6: Problem 6

Solve each logarithmic equation. Express irrational solutions in exact form. $$ \log (x+6)=1 $$

Short Answer

Expert verified
The solution is \( x = 4. \)

Step by step solution

01

Understand the Logarithmic Equation

We are given the equation \ \ \ \ \( \log (x+6)=1 \) . This is a logarithmic equation in terms of \( x \), with base 10 assumed, since no base is specified (common logarithm).
02

Rewrite the Equation in Exponential Form

Recall that the logarithmic equation \ \ \( \log_b y = x \) \) can be rewritten in exponential form as \( \ y = b^x \). Here, \( \( \log (x+6) = 1 \ \) means \ \ \( \( x+6 = 10^1 \) \).
03

Solve for \( x \)

Simplify the equation from Step 2: \ \ \[ x + 6 = 10^1 \] \[ x + 6 = 10 \] \[ x = 10 - 6 \] \ \ Therefore, \ \ \ x = 4. \
04

Verify the Solution

Substitute \( x = 4 \) back into the original equation to verify: \ \ \[ \ \ log (4 + 6) = \ log (10) = 1 \] \ \ Since this is correct, the solution is verified.

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Key Concepts

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

Logarithms
A logarithm is a mathematical function that helps us solve equations where the unknown variable is an exponent. The function tells us how many times we need to multiply a base number to get a certain value. For instance, in the equation \( \text{log}_b(y) = x \) \, the logarithm represents the power \( \x \) to which the base \( \b \) must be raised to produce \( \y \). In simpler terms, if \( \b = 10 \), then \( \text{log}(100) = 2 \) because 10 must be raised to the power of 2 to get 100. Logarithms have several properties, including:
  • \( \text{log}_b(m \times n) = \text{log}_b(m) + \text{log}_b(n) \)
  • \( \text{log}_b(\( \frac{m}{n} \) = \text{log}_b(m) - \text{log}_b(n) \)
  • \( \text{log}_b(m^k) = k \text{log}_b(m) \)
Exponential Form
Converting a logarithmic equation to its exponential form can make solving it easier. The general rule for this conversion is: \( \text{log}_b(y) = x \) can be rewritten as \( \y = b^x \). For the problem \(\text{log}(x+6)=1\) \, assume the base is 10 (if the base is not mentioned). Rewriting it in exponential form gives \( \x+6=10^1\) \. This conversion allows us to solve for \( \x \) more straightforwardly. It's crucial to understand this form because it is easier to work with exponentials for many logarithmic problems. Remember, whenever dealing with logarithms, the exponential form is often your best tool for simplification.
Solving Equations
Solving logarithmic equations involves a few systematic steps. For the given problem, \( \text{log}(x+6)=1\):
  • First, understand that this is a common logarithm, implying base 10.
  • Convert the logarithmic form into its exponential form: \ ( x+6 = 10^1). \
  • From the exponential form, simplify to \ ( x+6=10) \, and solve for \( x.\) \ This simplifies to \ (x=4). \
  • Lastly, always verify your solution by substituting back into the original equation and ensuring both sides are equal. Here, substituting \( x=4 \) back gives \( \text{log}(4+6) = \text{log}(10) = 1\).
Verification ensures that the solution not only meets mathematical correctness but eliminates potential calculation errors. Following these steps can make solving logarithmic equations much more efficient.

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Most popular questions from this chapter

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