What is the solution of the equation x log 10 (10/3) + log 10 3 = log 10 (2 + 3 x) + x?
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Solving the Logarithmic Equation for x
This explanation details the process of finding the value of the variable x in the given logarithmic equation. We will simplify the equation and check the potential solutions.
Understanding the Equation and Logarithm Rules
We are asked to solve the following equation involving base-10 logarithms:
\[ x \log_{10} \left( \frac{10}{3} \right) + \log_{10} 3 = \log_{10} (2 + 3 x) + x \]
To solve this problem, we will use key properties of logarithms:
- Quotient Rule: \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \)
- Power Rule: \( \log_b (M^k) = k \log_b M \)
- Base Identity: \( \log_{10} 10 = 1 \)
These rules help us manipulate the equation into a simpler form to find the value of x.
Step-by-Step Simplification of the Logarithmic Equation
Follow these steps to simplify the equation:
- First, simplify the term \( \log_{10}(10/3) \). Using the quotient rule, this is \( \log_{10} 10 - \log_{10} 3 \).
- Since \( \log_{10} 10 = 1 \), the term simplifies to \( 1 - \log_{10} 3 \).
- Substitute this back into the original equation: \[ x (1 - \log_{10} 3) + \log_{10} 3 = \log_{10} (2 + 3 x) + x \]
- Distribute x on the left side: \[ x - x \log_{10} 3 + \log_{10} 3 = \log_{10} (2 + 3 x) + x \]
- Subtract x from both sides of the equation to isolate the logarithm terms: \[ - x \log_{10} 3 + \log_{10} 3 = \log_{10} (2 + 3 x) \]
- Rearrange the terms to group the logarithms together: \[ \log_{10} 3 - x \log_{10} 3 = \log_{10} (2 + 3 x) \]
- Factor out \( \log_{10} 3 \) from the terms on the left side: \[ \log_{10} 3 (1 - x) = \log_{10} (2 + 3 x) \]
- Apply the power rule (\( k \log_b M = \log_b M^k \)) to the left side, moving the \( (1-x) \) factor into the logarithm: \[ \log_{10} (3^{1-x}) = \log_{10} (2 + 3 x) \]
- Since the logarithms on both sides have the same base (base 10) and are equal, their arguments must also be equal: \[ 3^{1-x} = 2 + 3x \]
The problem is now simplified to solving the exponential equation \( 3^{1-x} = 2 + 3x \). We can test the given options to see which one satisfies this condition.
Verification of Option 4 (x = 0)
Let's check if \( x = 0 \) is a valid solution. First, ensure that substituting \( x=0 \) into the original equation does not lead to undefined terms (like the logarithm of zero or a negative number).
- The term \( \log_{10} (10/3) \) is valid because \( 10/3 > 0 \).
- The term \( \log_{10} 3 \) is valid because \( 3 > 0 \).
- The term \( \log_{10} (2 + 3x) \) becomes \( \log_{10} (2 + 3(0)) = \log_{10} 2 \). This is valid because \( 2 > 0 \).
Now, substitute \( x = 0 \) into the original equation to check for equality:
Left Hand Side (LHS):
\[ LHS = 0 \times \log_{10} \left( \frac{10}{3} \right) + \log_{10} 3 \]
\[ LHS = 0 + \log_{10} 3 = \log_{10} 3 \]
Right Hand Side (RHS):
\[ RHS = \log_{10} (2 + 3 \times 0) + 0 \]
\[ RHS = \log_{10} 2 + 0 = \log_{10} 2 \]
Comparing the calculated LHS and RHS:
\[ \log_{10} 3 \neq \log_{10} 2 \]
This comparison shows that \( x = 0 \) does not satisfy the original equation.
We can also check \( x = 0 \) in the simplified form \( 3^{1-x} = 2 + 3x \):
Substitute \( x = 0 \):
LHS: \( 3^{1-0} = 3^1 = 3 \)
RHS: \( 2 + 3(0) = 2 + 0 = 2 \)
Since \( 3 \neq 2 \), the value \( x=0 \) does not satisfy the simplified equation either.
Summary of the Solution Process
The process involved applying logarithm rules to simplify the initial equation down to \( 3^{1-x} = 2 + 3x \). We then verified a potential solution, \( x=0 \), by substituting it back into both the original and simplified equations. The verification step is crucial for ensuring the accuracy of any proposed solution to a logarithmic or exponential equation.
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