In Exercises 89–102, determine whether each equation is true or f...

Question

In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement.

ln(8x^3) = 3 ln (2x)

Accepted Answer

Hello. Today we're going to be verifying whether the given equation is true or false and if it is false then we're going to verify whether the given changes are true. So the equation given to us is the natural log of 81 times a. To the fourth is equal to four times the natural log of three a. Now 81 is a pretty big number to work with but we can rewrite 81 as three to the power of four. So rewriting the left hand side is going to give us the natural log of three to the power four times eight to the power of four is equal to four times the natural log of three. A. Now recall that there is an exponential rule that states that if you have eight times B raised to the power of X. You can just go ahead and distribute this exponent to each term giving you a to the power of X times B to the power of X. And we can go ahead and apply this exponential rule to the inside arguments of the logarithms because we have three to the power four and eight to the power for. So if we utilize this property going from right to left, we can rewrite the left hand side as the natural log of three times A. To the power of four. And that is going to equal to four times the natural log of three A. Now we can go ahead and utilize the exponential rule for log for natural logarithms. And that states that if you have the natural log of A to the power of B, this is equal to B times the natural log of A. Here are B is going to be four, and R. A. Is going to be three A. For the left hand side. So utilizing this rule on the left hand side is going to give us four times the natural log of three A. And that is going to equal to four times the natural log of three a. And that verifies that the original equation is true. Thus the answer to this problem is going to be a. So I hope this video helps you and understanding how to approach this type of problem, and I'll go ahead and see you all in the next video.

In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. ln(8x^3) = 3 ln (2x)

Key Concepts

Properties of Logarithms

Natural Logarithm (ln)

Equation Verification

Watch next