SOLUTION: Help express this as a single logarithm and simplify if possible. log(2)64

SOLUTION: Help express this as a single logarithm and simplify if possible. log(2)64 - log(4)4

Question 420975: Help express this as a single logarithm and simplify if possible.
log(2)64 - log(4)4
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

A problem like this is fairly easy once you understand what logarithms are. The idea behind logarithms is that is it possible to raise any positive number (excpet 1) to the "right" power and get any other positive number as a result. For example:

A 3, raised to the right power, can result in a 81.
A 1/2, raised to the right power, can result in an 8.
A 2000, raised to the right power, can result in a 17.

So what are these "right powers"? Well, some of them are relatively easy to figure out. The exponents for the first two examples above, can be figured out without a lot of effort. A 3 raised to the 4th power results in an 81. And a 1/2 raised to the -3 power results in an 8.

Many of these "right powers" are difficult to find. But even though they may be hard to find, the idea behind logarithms says that they do exist.

A notation is used to express these exponents that is difficult to understand at first. This notation is used for both the easy-to-find and the difficult-to-find exponents. It looks like this:

This notation is used to represent "the exponent for 'a' that results in 'b'". The "a" is called the base of the logarithm. The "b" is called the argument of the logarithm. Any positive number (except 1) can be used for the base and the argument can be any positive number (including 1). Zeros or negative numbers are not allowed for either the base or the argument of a logarithm!

Using the notation to express the exponents described in the three examples above we would use:

and

Now let's look at each of the logarithms in your problem:

This represents the exponent for 2 that results in 64. If we can figure out what this exponent is (IOW: if we can figure out what power of 2 is 64), then we can simply replace this logarithm with that exponent. After trying some powers of 2 it should not take you long to figure out that the exponent for 2 that results in 64 is 6. So we can replace this logarithm with a 6.

This represents the exponent for 4 that results in 4. This should be ridiculously obvious. The exponent is 1!.

So both of your logarithms are the kind that are easily found. And replacing your logarithms with their values we get:
6 - 1
which simplifies to 5.

Here are some important ideas and properties to understand when working with logarithms:

Logarithms are exponents.
By the definition of what means:
By the definition of what means:
The base of a logarithm can be changed using:

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