Question

Simplify the logarithmic expression. 3log12(2) + 13log12(8) - log12(4).

a) -2
b) 7
c) 6
d) 5

Answer 1

To simplify the given logarithmic expression, we can use the properties of logarithms such as the property that states the logarithm of the number resulting from the division of two numbers is the difference between the logarithms of the two numbers. By applying these properties, we can simplify the expression step by step.

Step-by-step explanation:

To simplify the given logarithmic expression, we can use the properties of logarithms. The property that states the logarithm of the number resulting from the division of two numbers is the difference between the logarithms of the two numbers can be applied in this case.

1. Consider the first term, 3log12(2). Since the base is the same, we can multiply the coefficient by the exponent: 3log12(2) = log12(2^3) = log12(8).
2. Next, consider the second term, 13log12(8). Again, we use the same property: 13log12(8) = log12(8^13).
3. Finally, consider the third term, -log12(4). This term can be simplified to log12(1/4) since dividing by 4 is the same as multiplying by its reciprocal.

Combining the simplified terms, we have log12(8) + log12(8^13) - log12(1/4). Applying the property of addition, log12(8) + log12(8^13) can be combined into a single logarithm: log12(8 × 8^13).

Now, we can use the property that states the logarithm of the product of two numbers is the sum of the logarithms of the individual numbers: log12(8 × 8^13) = log12(8^(1+13)).

Finally, simplifying further, we have log12(8^14) = 14log12(8).

Therefore, the simplified expression is 14log12(8) - log12(1/4), which can also be written as log12(8^14) - log12(1/4).