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.

x log 10^x = x^2

Accepted Answer

Hello. Today we're going to be verifying whether the given equation is true or false and if it is false we need to verify whether the given changes are true. So the equation given to us is n squared times the quantity, the logo rhythm of base two of two to the power of N is equal to N. Cute. Now one thing that we can go ahead and do is get rid of this exponent inside the logarithms argument and we can do that using the exponential rule for logarithms. And that states that if you have the logarithms of A to the power of B you can rewrite this as B times the logarithms of A. So you can go ahead and bring this end in front of the logarithms as a product and doing this is going to give us the end squared times the quantity and times the logarithms of base two of two and that is going to equal to end cubed. Now we can go ahead and combine the coefficients together because we can multiply and squared with N and n squared times N is going to give us and cubed. So what we are left with is n cubed times the logarithms of base two of two and that is going to equal to n. Cute. Now there is one more logarithmic property that we can apply here. There is a logarithmic property that states that if you have the logarithms whose base matches its argument. Then this can be rewritten as one and here the basis to and the argument is to. So this logarithms base an argument, match each other so you can go ahead and rewrite the logarithms of base two of two as one. And what you're left with is n cubed times one, and n cubed times one is just going to give us n cubed. So what we are left with is n cubed is equal to n cubed. And that means that this is going to be a true statement. So the answer to this problem is going to be a so I hope this video helps you in 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. x log 10^x = x^2

Key Concepts

Logarithmic Properties

Exponential Functions

Equation Validity

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