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Chapter 6: Problem 10

Solve each logarithmic equation. Express irrational solutions in exact form. $$ \log _{5}(2 x+3)=\log _{5} 3 $$

Short Answer

Expert verified
x = 0

Step by step solution

01

- Set the Logarithmic Arguments Equal

Since the bases of the logarithms on both sides of the equation are the same, you can set the arguments equal to each other:\[2x + 3 = 3\]
02

- Solve for x

Now solve the equation for x by isolating the variable. First subtract 3 from both sides:\[2x + 3 - 3 = 3 - 3\]This simplifies to:\[2x = 0\]Next, divide both sides by 2:\[x = 0\]
03

- Verify the Solution

Substitute x = 0 back into the original equation to verify the solution:\[\log_{5}(2(0)+3) = \log_{5}3\]This simplifies to:\[\log_{5}3 = \log_{5}3\]Since both sides of the equation are equal, the solution x = 0 is verified.

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Key Concepts

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

solving logarithmic equations
To solve a logarithmic equation like \( \log _{5}(2 x+3)=\log _{5} 3 \), follow these steps:
  • Identify the logarithms. In this example, we have \( \log _{5} \) on both sides with the arguments \( 2 x+3 \) and 3.
  • Because the bases are the same, set the arguments equal to each other: \( 2 x + 3 = 3 \)
  • Solve for the variable \( x \). Subtract 3 from both sides: \( 2 x + 3 - 3 = 3 - 3 \) which simplifies to \( 2 x = 0 \).
  • Divide both sides by 2: \( x = 0 \).
  • Always verify the solution by substituting \( x \) back into the original equation. Here, putting \( x = 0 \) back gives \( \log_{5}(2(0)+3) = \log_{5}3 \), confirming that both sides are equal.
logarithms
Understanding logarithms is key to solving these equations. A logarithm answers the question: 'To what exponent must the base be raised, to obtain a given number?' We write this as: \( \log_b(a) = c \) means \( b^c = a \). In our example, base \( b \) is 5.
algebra
Basic algebra steps are essential in solving logarithmic equations. After setting the arguments equal, isolate the variable by performing operations like addition, subtraction, multiplication and division. For \(2x + 3 = 3\), subtract 3 from both sides to get \(2x = 0\), then divide by 2 to find \(x = 0\).

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