Problem 123 - Algebra and Trigonometry

Question

Explain the differences between solving \(\log _{3}(x-1)=4\) and \(\log _{3}(x-1)=\log _{3} 4\).

Step-by-Step Solution

Answer

The solutions are \(x=82\) for the equation \(\log _{3}(x-1)=4\) and \(x=5\) for the equation \(\log _{3}(x-1)=\log_{3}4\) respectively. For the first equation, the logarithm had to be converted into an exponential form and subsequently, 'x' was isolated. Whereas in the second case, the property of equal logarithms was used to simplify the equation and solve it.

Step 1: Convert logarithmic equation to exponential form

For the equation \(\log _{3}(x-1)=4\), applying the exponent property of logarithms, where \(\log_b y = x\) can be rewritten as \(b^x = y\), the equation can be rewritten as \(3^4=x-1\).

Step 2: Solve for x

Solving for x requires isolating x. By adding 1 to both sides of the equation \(3^4=x-1\), the equation becomes \(3^4+1=x\). Evaluating \(3^4\) equals 81, so adding 1 to 81 gives \(x=82\).

Step 3: Use the property of equal logarithms

For the equation \(\log _{3}(x-1)=\log _{3} 4\), employ the property of logarithms that states if two logarithms with the same base are equal, then their arguments must also be equal. Therefore, \(x-1=4\).

Step 4: Solve for x

Solving for x in the equation \(x-1=4\), requires adding 1 to both sides of the equation to isolate x, resulting in \(x=4+1=5\).