Solve for $x$.
$\log _3(2 x-9)=1$
Answer (1)
Rewrite the logarithmic equation in exponential form: 2 x − 9 = 3 1 .
Solve the resulting linear equation for x : 2 x − 9 = 3 .
Add 9 to both sides: 2 x = 3 + 9 , which simplifies to 2 x = 12 .
Divide both sides by 2: x = 2 12 , so x = 6 . Therefore, the solution is 6 .
Explanation
Understanding the Problem We are given the equation lo g 3 ( 2 x − 9 ) = 1 and we need to solve for x .
Converting to Exponential Form To solve for x , we first rewrite the logarithmic equation in exponential form. The equation lo g 3 ( 2 x − 9 ) = 1 is equivalent to 3 1 = 2 x − 9 .
Isolating x Now we have the equation 2 x − 9 = 3 . We want to isolate x , so we add 9 to both sides of the equation: 2 x − 9 + 9 = 3 + 9 , which simplifies to 2 x = 12 .
Solving for x Next, we divide both sides of the equation by 2 to solve for x : 2 2 x = 2 12 , which simplifies to x = 6 .
Final Answer Therefore, the solution to the equation lo g 3 ( 2 x − 9 ) = 1 is x = 6 .
Examples
Logarithmic equations are used in various fields, such as calculating the magnitude of earthquakes on the Richter scale, determining the acidity or alkalinity (pH) of a solution in chemistry, and modeling population growth or decay in biology. Understanding how to solve logarithmic equations allows us to analyze and interpret data in these real-world scenarios. For example, if we know the magnitude of an earthquake is 5 on the Richter scale, we can use logarithmic equations to determine the amplitude of the seismic waves. Similarly, in chemistry, we can use logarithmic equations to calculate the concentration of hydrogen ions in a solution given its pH value.
