Solve each equation. Give solutions in exact form. See Examples 5...

Question

Solve each equation. Give solutions in exact form. See Examples 5–9. log(2 - x) = 0.5

Accepted Answer

Hey, everyone here we are asked to solve the logarithmic equation for X and provide an exact solution here. The given equation is long of 19 minus X is equal to 0.25. Here we have four answer choice options. Answer A, the solution set of the square root of 10 plus 19. Answer B the square root of 10 minus 19. Answer C the solution set of 19 plus the fourth route of 10 and answer D the solution set of minus the fourth route of 10. So here again, our given equation is log of 19 minus X is equal to 0.25. And to solve this given equation for X, we need to recall that log space A of X is equal to Y has the equivalent form where X is equal to A, to the power of Y. So now we just want to compare our first formula where we, again, we have log base A of X is equal to Y to our given equation to identify the values for each of our variables. So we can start with the value for a. Well, here we see in our given equation that we just have log well log without an identified base implies log base 10. Therefore, our value for a, in this case is going to be 10. Now for our value or in this case expression for X, we have 19 minus X. And lastly, our value for Y is just the constant on the right side of our given equation. So Y is going to be 0.25. So now with each of our variables clearly identified, we can rewrite our given equation in the second form where X is equal to a to the power of Y. So using this form, we have 19-1 is equal to 10 to the power of 0.25. Well, rewriting 10 to the power of 0.25 is equivalent to 10 to the power of 1/4 and 10 to the power of 1/ is equivalent to the fourth route of 10. So now we want to continue just solving for X. So again, we have 19 minus X is equal to the fourth route of 10. And now solving for X, we first need to subtract 19 from both sides. So we have negative X is equal to negative 19 plus the fourth route of 10. Now just getting rid of the coefficient of X, we need to divide both sides by negative one. And we see we have X is equal to 19 minus route of 10. So with this value for X, we can write it as a solution set where we have a solution set of 19 minus the fourth root of 10. So this is our final answer here, which leaves us with answer choice D where again, we have the solution set of 19 minus the fourth route of of 10. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Solve each equation. Give solutions in exact form. See Examples 5–9. log(2 - x) = 0.5

Key Concepts

Logarithmic Functions

Properties of Logarithms

Solving Exponential Equations

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