Solve each equation for the indicated variable. Use logarithms wi...

Question

Solve each equation for the indicated variable. Use logarithms with the appropriate bases. See Example 10. I = E/R (1- e^(-(Rt)/2), for t

Accepted Answer

Hey, everyone here we are asked to solve four K in the given expression by making use of algorithms with proper bases. Here we are given the expression Q is equal to a divided by two B times the quantity of one minus E raised to the power of negative P K divided by three. Here we have four answer choice options. Answer A K is equal to three divided by P time. The natural log of A divided by the quantity of A minus two Q B answer B K is equal to P divided by three times the natural log of A divided by the quantity of A minus two Q B. Answer C K is equal to three divided by P times Z natural log of A divided by the quantity of A plus two Q B. And answer D K is equal to P divided by three times Z natural log of A divided by the quantity of A plus two Q B. So now looking back at our given expression in order to solve four K, we first want to manipulate our expression such that we isolate E since it contains a power including the variable K. So to isolate B, we first want to get rid of this coefficient by dividing both sides by the coefficient which is A divided by two B. So we divide both sides by A divided by two B. And now we can rewrite our expression as two Q B divided by A is equal to one minus E raised to the power of negative P K divided by three. And now we can just move this one from the right side to the left side of the equation by simply subtracting one from both sides. And so now we will have two Q B divided by A minus one is equal to negative E raised to the power of negative P K divided by three. And so now we want to get E to be positive. So we can just multiply both sides of the equation by negative one. And so in doing this multiplying both sides by negative one, we get one U B divided by A is equal to E raised to the power of negative P K divided by three. And now we just want to simplify the left side of our equation by combining both of the terms. And so combining the terms, we have A minus two Q B all divided by A is equal to E raised to the power of negative P K divided by three. And now we want to manipulate this current equation such that we get K to be outside of this exponent. So that means we need to take the natural log of both sides. So taking the natural log, we have the natural log of A minus two Q B divided by A is equal to the natural log of E raised to the power of negative P K divided by three. And now, in order to get K to be outside of this exponent of E, we need to recall the logarithmic property which states that power of X is equivalent to X times the natural log of A. So now utilizing this property, we can rewrite the right side of our equation. So we have the natural log of A minus two Q B all divided by A is equal to negative P K divided by three times the natural log of E. And now we can continue to simplify the right side of our equation by recalling that the natural log of E is equivalent to one. So since it is equivalent to one, we simply have negative P K divided by three is equal to the natural log of A minus two Q B all divided by A. And so now again, we want K to be alone because we are solving four K. So that means the next step is to simply divide both sides by negative P divided by three. And so in doing this, we have isolated K. So our expression is now K is equal to three negative three divided by P times a natural log of A minus two Q B all divided by A. And so now we just want to simplify our expression further by getting rid of the negative in our coefficient. So to get rid of the negative, we simply recall the same logarithmic property from earlier where we have that the natural log of A to the power of X is equivalent to X times the natural log of A. And we can utilize this property to move this negative sign to be a power of the natural log. So now utilizing this property, we have the expression K is equal to now positive three divided by P times the natural log of A minus two Q B all divided by A now raised to the power of negative one. And now with this power of negative one, we can simplify our expression further by rewriting the expression contained within the natural log. So our final expression becomes K is equal to three divided by P times the natural log of A divided by the quantity of A minus two Q B. So with our final simplified expression for K, we have finished solving our original expression for K. And so we are left here with answer choice A where again, we have K is equal to three divided by P times the natural log of A divided by the quantity of A minus two Q B. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Solve each equation for the indicated variable. Use logarithms with the appropriate bases. See Example 10. I = E/R (1- e^(-(Rt)/2), for t

Key Concepts

Exponential Functions

Logarithms

Algebraic Manipulation

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