SOLUTION: Use properties of logarithms to find the exact value of each expression. 5^log base 5^7

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Question 608851: Use properties of logarithms to find the exact value of each expression.
5^log base 5^7
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
5%5Elog%285%2C+%287%29%29
Once you really get what logarithms are, a problem like this is almost as easy as 1 + 1. So first let's look at the exponent:
log%285%2C+%287%29%29
and try to understand what it represents.

The idea behind logarithms is that it is possible to take any positive number (except 1) and, if you raise it to the right power, get any positive number as a result. For example, it is possible to raise 4 to some power and get 16. In fact you should know what the "right power" is: 2. It also possible to raise 4 to the "right power" and get 17! This exponent is not well-known but it exists. Similarly we can raise 12000 to some power and get 1/3, raise 51.3 to some power and get 1000, etc.

In general, logarithms are exponents. log%28a%2C+%28p%29%29 is how we express the "right exponent" for a to get p. For example:
log%284%2C+%2816%29%29 represents the exponent for 4 that results in 16.
log%284%2C+%2817%29%29 represents the exponent for 4 that results in 17.
log%2812000%2C+%281%2F3%29%29 represents the exponent for 12000 that results in 1/3.
log%2851.3%2C+%281000%29%29 represents the exponent for 51.3 that results in 1000.
etc.

Some exponents are well-known:
log%284%2C+%2816%29%29+=+2 since 2 is the exponent for 4 that results in 16
log%282%2C+%2864%29%29+=+6 since 6 is the exponent for 2 that results in 64
log%2825.93%2C+%281%29%29+=+0 since any number raised to the 0 power is 1
log%2825%2C+%285%29%29+=+1%2F2 since 5 is the square root of 25 and an exponent of 1/2 means square root.
etc.

Many logarithms are not well-known:
log%284%2C+%2817%29%29
log%2812000%2C+%281%2F3%29%29
log%2851.3%2C+%281000%29%29
etc.
For these, if we need to see a number for the exponent for some reason, we have to reach for our calculators.

Now let's look at your logarithm:
log%285%2C+%287%29%29
It represents the exponent for 5 that results in a 7. This is not a well-known... But look at where we see it! It's the exponent for a 5. So your full expression is "5 raised to the power for 5 which results in a 7". So:
5%5Elog%285%2C+%287%29%29+=+7!!!

In fact, the base number doesn't matter.
a%5Elog%28a%2C+%28p%29%29+=+p
no matter what "a" and "p" are (as longs as they are positive and a is not 1). It's like:
  1. Pick a positive number (not 1).
  2. Now raise it to a power that we know will result in "p" (another positive number).
  3. What answer do you get? "p", of course!