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The only way to find a base 2 logarithm without a calculator is to have an argument that is a (obvious) powers of 2. At the moment, none of the arguments are powers of two. So we will start by manipulating the logarithms in the hope of getting arguments that are powers of 2.
The only way to change arguments in logarithms are with one of the properties of logarithms:
The first two properties require that the logarithms have the base and that they have coefficients of 1. The third property allows us to "move" a coefficient into the argument as an exponent. Since we will be wanting to use one of the first two properties on your third logarithm, we will start by using the third property to "move" the coefficient (really the coefficient changes to a 1) in the argument:
which simplifies to:
Now, because of the "-" between them, we can use the second property above to combine the first two logarithms:
which simplifies as follows:
Next, because of the "+" between them, we can use the first of the properties above to combine the remaining logairhtms:
which simplifies to:
By using the properties to simplify the expression down to one logarithm, the argument changed into a power of 2. This we can find "by hand". Since , . So your expression simplifies to 2.
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