Overview
This lecture demonstrates how to solve a logarithmic equation by isolating the logarithm, converting to exponential form, and verifying the solution.
Solving the Logarithmic Equation
- Start with the equation: ( 2 \log_{10} x + 2 = 8 ).
- Subtract 2 from both sides: ( 2 \log_{10} x = 6 ).
- Divide both sides by 2: ( \log_{10} x = 3 ).
- Rewrite the logarithmic equation in exponential form: ( x = 10^3 ).
- Calculate the value: ( x = 1000 ).
Verifying the Solution
- Substitute ( x = 1000 ) into the original equation.
- ( 2 \log_{10} 1000 + 2 = 2 \times 3 + 2 = 6 + 2 = 8 ).
- Check that both sides are equal; thus, the solution is correct.
Understanding Logarithmic and Exponential Forms
- If ( \log_{10} x = 3 ), the equivalent exponential form is ( x = 10^3 ).
- Converting between logarithmic and exponential forms helps to isolate the unknown.
Key Terms & Definitions
- Logarithm (( \log_{10} x )) — The power to which 10 must be raised to get ( x ).
- Exponential Form — An expression where the variable is the exponent (e.g., ( x = 10^3 )).
Action Items / Next Steps
- Practice converting logarithmic equations to exponential form and solving for the unknown.
- Verify solutions by substituting values back into the original equation.