Expanding Log Expressions

log₂(8x⁵) = log₂(8) + log₂(x⁵)

When the instructions say to "expand", they mean that they've given me one log expression with lots of stuff inside it, and they want me to use the log rules to take the log apart into many separate log terms, each with only one thing inside its particular log. That is, they've given me one log with a complicated argument, and they want me to convert this to many logs, each with a simple argument.

In this case, I have a "2x" inside the log. Since "2x" is multiplication, I can take this expression apart, according to the Product Rule, and turn it into an addition outside the log:

There is nothing more I can do to either of these log expressions, so the answer they are looking for is:

I have division inside the log. According to the Quotient Rule, this can be split apart as subtraction outside the log, so:

The first term on the right-hand side of the above equation can be simplified to an exact value, by applying the basic definition of what a logarithm is. In this case, I'm using the fact that the power required on 4 to create 16 is 2; in other words, since 4² = 16, then:

Then the original expression expands fully as:

The exponent inside the log can be taken out front as a multiplier, so my answer is:

Okay; they've told me to "expand", so I know they're wanting me to take this one log apart, into many log terms.

I'll start with the division inside this log. The 5 is divided into the 8x⁴, so I'll apply the Quotient Rule and split the numerator and denominator apart by converting the one log containing division into two subtracted logs:

The second term above, with just a 5 inside, is as "expanded" as that term can get, because there's only just the one thing inside the log. And, because 5 is not a power of 2, there's no further expansion that I can do. So that part of this expansion is done; I'll just be carrying the "log(5)" along for the ride to the final answer.

In the other log term, though, there's still more than just one thing inside the log. In particular, I see that there's an exponent inside the log. However, I can't take the exponent out front yet, because that power is only on the x, not the 8. I have to remember that the rule says that I can only take the exponent out front if it is on everything inside the log. So I first need to isolate that part of the argument that has the power on it.

The 8 is multiplied onto the x⁴, so I can split the factors inside the log by using the Product Rule to convert this one log with multiplication into two logs that are added:

Since 8 is a power of 2 (namely, 2³), I can simplify the first log to an exact value. Because 2³ = 8, then log₂(8) = 3, so I get:

Okay; now I'm finished with the first term, too; I'm only left with the middle term to expand, with the exponent inside its log.

The variable x has the exponent (which is now on everything inside its log), so I can use the Power Rule to move the exponent out in front of the log as a multiplier:

Each log now finally contains only one thing, and the first log term has been simplified to an exact (as opposed to approximate decimal) numerical value, so this expression is fully expanded. Then my final answer is:

This is a gawd-awful mess! I will be taking extra care not to try to do anything in my head, nor to try to do too much all at once.

The first thing I see, inside the log, is that I've got one complicated expression that's divided by another complicated expression. To start my expansion, then, I'll split the division inside the log into subtraction of logs outside.

Inside each of the two log terms I've got now, I find multiplication. So my next step will be to take apart the multiplications inside as addition of logs outside.

To make sure I don't mess up my signs, I'll be sure to put grouping symbols around the results of each split.

Now I'll take the middle "minus" through the square brackets.

Inside all but the first term, I find exponents. Since each exponent is on the entire contents of its respective log, I can go straight to moving the powers inside to being multipliers outside.

Then my hand-in answer is: