Exponential and Logarithmic Equations solve the equation. Find the exact solution if possible. otherwise, use a calculator to approximate to two decimals. ln⁡(x−2)+ln⁡3=ln⁡(5x−7)

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Exponential and Logarithmic Equations solve the equation. Find the exact solution if possible. otherwise, use a calculator to approximate to two decimals.  ln⁡(x−2)+ln⁡3=ln⁡(5x−7)

Maciej Morrow · Accepted Answer

Step 1  Law of logarithm:  Consider m to be a positive number and m≠q1.  Again consider M and N to be any real numbers with M&amp;gt;0 and N&amp;gt;0.  The sum of logarithms of two numbers is equal to the logarithm of product of two numbers as,  logmM+logmN=logm(MN)  Step 2  The given exponential equation is,  1) ln⁡(x−2)+ln⁡3=ln⁡(5x−7)  The above logarithm equation can be comined from the laws of logarithm as,  ln⁡3(x−2)=ln⁡(5x−7)  3(x−2)=5x−7  3x−6=5x−7 Simplify the equation 3x−6=5x−7 as,  3x−5x=−7+6  −2x=−1  2x=1  x=12  Now substitute 12 for x in equation (1) as,  ln⁡(12−2)+ln⁡3=ln⁡(5(12)−7)  ln⁡(−32)+ln⁡3=ln⁡(52−7)  ln⁡(−32)+ln⁡3=ln⁡(5−2⋅72)  ln⁡(−32)+ln⁡3=ln⁡(−92)  The logarithm is defined only for real numbers.  Therefore, x=12 is not a solution of equation (1).  Conclusion:  Thus, there is no solution of the logarithm equation ln⁡(x−2)+ln⁡3=ln⁡(5x−7).

Solve the equation. Find the exact solution if possible \ln(x-2)+\ln3=\ln(5x-7)

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