SOLUTION: How do I rewrite the statment of log4(8) using the properties of logarithms and simplify if possible

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Question 594309: How do I rewrite the statment of log4(8) using the properties of logarithms and simplify if possible
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Logarithms can be calculated "by hand" if
  • Case 1: The argument is a known power of the base. In this case the answer is simply that power. For example, log%282%2C+%288%29%29+=+3 since 8 (the argument) is 2 (the base) to the 3rd power.
  • Case 2: The base is a known power of the argument. In this case use the change of base formula, log%28a%2C+%28p%29%29+=+log%28b%2C+%28p%29%29%2Flog%28b%2C+%28a%29%29, to convert the base to the value of the argument. For example, log%288%2C+%282%29%29. Change the base to 2 (the argument):
    log%288%2C+%282%29%29+=+log%282%2C+%282%29%29%2Flog%282%2C+%288%29%29
    Both of the logs on the right are "case 1" logs (above). So
    log%288%2C+%282%29%29+=+log%282%2C+%282%29%29%2Flog%282%2C+%288%29%29+=+1%2F3
  • Case 3: Both the argument and the base are known powers of some third number. In this case use the change of base formula to change the base to that third number. For example. log%2832%2C+%2864%29%29. 32 and 64 are not well-known powers of each other. But they are both powers of 2. So we use the change of base formula to change to base 2 logs:
    log%2832%2C+%2864%29%29+=+log%282%2C+%2864%29%29%2Flog%282%2C+%2832%29%29
    Since 2%5E6+=+64 and 2%5E5+=+32 the logs on the right become 6 and 5 respectively:
    log%2832%2C+%2864%29%29+=+log%282%2C+%2864%29%29%2Flog%282%2C+%2832%29%29+=+6%2F5

For log%284%2C+%288%29%29 you will need to use the change of base formula to simplify this (unless you are clever enough to figure out what power of 4 results in an 8). Since both 4 and 8 are powers of 2 we will use the change of base formula to change the base to 2:
log%284%2C%288%29%29+=+log%282%2C%288%29%29%2Flog%282%2C%284%29%29+=+3%2F2+=+1%261%2F2

In case the change of base formula is not considered a "property" of logarithms, here is an alternate solution. This solution uses the property:
log%28a%2C%28p%29%29+%2B+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Aq%29%29
Normally we use this property, from left to right, to "add" two logarithms. But most properties in Math can be used in both directions and here we will use it from right to left to split the log of a product into the sum of two logs.

You might wonder, "But 8 is not a product!?" You're right. It isn't ... not yet at least. We will start by factoring 8 (IOW turn it into a product) using factors whose logs are easier to find:
log%284%2C%288%29%29+=+log%284%2C%284%2A2%29%29
Now we can use the property:

log%284%2C%284%29%29 is about as easy as it gets with logarithms. For log%284%2C%282%29%29 we will have to work a little harder. We have to figure out what power of 4 is equal to 2. Since 2 is the square root of 4 and since an exponent of 1/2 represents a square root we now know what this logarithm is. Putting this all together we get: