Solve the equation for $x$.
$\log _3(6 x-5)=1$
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. $x=$ $\square$ (Simplify your answer.)
B. There is no solution.
Answer (1)
Convert the logarithmic equation to exponential form: 3 1 = 6 x − 5 .
Solve the resulting linear equation for x : 6 x = 8 .
Simplify the fraction: x = 3 4 .
Check if the solution is valid: 0"> 6 ( 3 4 ) − 5 = 3 > 0 . The final answer is 3 4 .
Explanation
Understanding the Problem We are given the equation lo g 3 ( 6 x − 5 ) = 1 and we need to solve for x . This is a logarithmic equation, and to solve it, we will convert it to its equivalent exponential form.
Converting to Exponential Form To convert the logarithmic equation to exponential form, we use the definition of logarithms: lo g b ( a ) = c is equivalent to b c = a . In our case, b = 3 , a = 6 x − 5 , and c = 1 . Therefore, the exponential form of the equation is 3 1 = 6 x − 5 .
Isolating x Now we have the equation 3 = 6 x − 5 . We need to isolate x . First, we add 5 to both sides of the equation: 3 + 5 = 6 x − 5 + 5 8 = 6 x
Solving for x Next, we divide both sides of the equation by 6 to solve for x : 6 8 = 6 6 x x = 6 8
Simplifying the Fraction We can simplify the fraction 6 8 by dividing both the numerator and the denominator by their greatest common divisor, which is 2: x = 6 ÷ 2 8 ÷ 2 x = 3 4
Checking the Solution Now we need to check if this solution is valid by plugging it back into the original equation. The argument of the logarithm, 6 x − 5 , must be greater than 0. So, we need to check if 0"> 6 ( 3 4 ) − 5 > 0 : 6 ( 3 4 ) − 5 = 3 24 − 5 = 8 − 5 = 3 Since 0"> 3 > 0 , the solution x = 3 4 is valid.
Final Answer Therefore, the solution to the equation lo g 3 ( 6 x − 5 ) = 1 is x = 3 4 .
Examples
Logarithmic equations are used in various fields, such as calculating the magnitude of earthquakes on the Richter scale, measuring sound intensity in decibels, and determining the pH of a solution in chemistry. For instance, if we know the hydrogen ion concentration of a solution is 1 0 − 7 M, we can use logarithms to find the pH: p H = − lo g 10 ( 1 0 − 7 ) = 7 . This shows how logarithms simplify calculations and provide meaningful interpretations in scientific contexts.
