SOLUTION: lob(base 64)x+log(base 8)x+log(base 2)x=9 Write the equation as a single logarithm. Express powers as factors. Show all work

Question 638684: lob(base 64)x+log(base 8)x+log(base 2)x=9 Write the equation as a single logarithm. Express powers as factors. Show all work
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

We want the equation in the form:
log(expression) = number
So we need to combine the logs into one. And whether you add the logs or use properties to combine them, the bases of the logs need to be the same. So our first task is to use the change of base formula, , to change the bases so that they are all the same.

With a little investigation we should be able to find that both 64 and 8 are powers of 2.

Since we know what powers of 2 result in 64 and 8 we can replace the denominators:

All these terms are like terms so we can add them. But before that let's get rid of the fractions. Multiplying the the lowest common denominator, 6, we get:

Now we can add the logs. Exactly like q + 2q + 6q = 9q,

becomes:

Dividing by 9 we get:

We now have the desired
log(expression) = number
form. The next step is to rewrite this in exponential form. In general is equivalent to . Using this pattern on our equation we get:

which simplifies to:

You should always check solutions to this kind of equation. You must make sure that all bases and arguments of all logarithms remain positive. Use the original equation to check:

Checking x = 64:

We can already see that the bases and arguments are all positive. Our solution passes the required part of the check. (You are welcome to finish the check to see if we made a mistake.)
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