Solve for $x$.
$\log _4(-3 x+8)=1$
Answer (1)
Rewrite the logarithmic equation in exponential form: 4 1 = − 3 x + 8 .
Solve the linear equation for x : 4 = − 3 x + 8 .
Isolate x : − 4 = − 3 x .
Divide by -3: x = 3 4 .
3 4
Explanation
Understanding the problem We are given the equation lo g 4 ( − 3 x + 8 ) = 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 4 ( − 3 x + 8 ) = 1 is equivalent to 4 1 = − 3 x + 8 .
Setting up the linear equation Now we have a linear equation 4 = − 3 x + 8 . We need to isolate x .
Isolating x Subtract 8 from both sides of the equation: 4 − 8 = − 3 x , which simplifies to − 4 = − 3 x .
Solving for x Divide both sides by -3 to solve for x : x = − 3 − 4 .
Simplifying the solution Simplify the fraction: x = 3 4 .
Checking the solution Now, we need to check if this solution is valid by substituting x = 3 4 back into the original equation. We need to ensure that 0"> − 3 x + 8 > 0 because the logarithm of a non-positive number is undefined.
Substituting x = 3 4 , we get − 3 ( 3 4 ) + 8 = − 4 + 8 = 4 . Since 0"> 4 > 0 , the solution is valid.
Final Answer Therefore, the solution to the equation lo g 4 ( − 3 x + 8 ) = 1 is x = 3 4 .
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 in biology. For example, if we know the intensity of an earthquake is 1000 times greater than the smallest detectable wave, we can use logarithms to find its magnitude on the Richter scale: M = lo g 10 ( 1000 ) = 3 . This shows how logarithms help us to work with very large or very small numbers in a more manageable way.
