Solve each logarithmic equation in Exercises 49–92. Be sure to re...

Question

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 2 log x−log 7=log 112

Accepted Answer

Welcome back. I am so glad you're here we are given this log rhythmic equation three times blog of X minus log of three equals the log of 243 and we need to solve for X. Well it's inside this log we've got to do a little bit of simplifying and rearranging in order to get that out of there. So let's first recall from previous lessons on our properties of logarithms that when we have a number in front of being multiplied by our log, that is also an exponent of our log of whatever. That's the log of. So in this case it's an exponents of X. We can rewrite that as log of X cubed -63 equals log of 243. And now we recall from condensing our logarithms that when we have a log of something minus a log of something, we can rewrite that as the log of those to some things divided by each other. So the first one goes in the numerator X cubed divided by the second one goes in the denominator three that is still equal to the log of 243. Now we recall from the 1 to 1 property of logarithms that the log of something equal to the log of something else tells us that the something is equal to the something else. So exc cubed divided by three is equal to 243 and ye thank you 1 to 1 property of logarithms. Now we have rewritten that without the logs we can solve for X. Let's multiply both sides by three. So we'll have on the left X cubed and on the right 243 times three is 729. Let's take the cubed root of both sides to get that X by itself. And now we have X equals the cubed root of 729 is nine. We have X equals nine. Well we've got a solution but we do need to check because there are domain restrictions for logs. We need to make sure that what is inside of our log here is a positive value. So we'll plug nine in there and yeah nine cubed. That's a positive value. So we look at our answer choices and we go with answer choice B. Well done. We'll catch you on the next one.

Key Concepts

Logarithmic Properties

Domain of Logarithmic Functions

Decimal Approximation

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