
The product of lies between two successive integers which are _______ and _______.
Answer
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Hint: We change the base for logarithms involved in the given question to new base using the base change formula and then multiply. We use logarithmic identity of product and logarithmic identity of quotient to proceed and use the logarithmic table to evaluate.
Complete step-by-step solution
We know that the logarithm is the inverse operation to exponentiation. That means the logarithm of a given number is the exponent to which another fixed number, the base must be raised, to produce that number , which means if then the logarithm denoted as log and calculated as
Here are real numbers subjected to the condition that the argument of logarithm is always a positive number and is a positive number excluding 1. If we want to change the base of the logarithms to new base say then we can do it using following formula,
We multiply the three given logarithms in the question and have
We change the base of each logarithm to base using the base change formula with in the first logarithm, with in the second logarithm and with in third logarithm. We have,
We use the logarithmic identity of product for and logarithmic identity of quotient for in the above step and proceed to have,
We check the logarithm table and put logarithmic value in the above step to have,
So the product of logarithms is approximately and the integers between which are -1 and -2.
Note: We note that if we logarithm with base 10 is called common logarithm and we have changed the base to 10 because we know about common logarithm, We note that base cannot be 1 or negative. If we do not want to use logarithm table we can use logarithmic identities inequality .
Complete step-by-step solution
We know that the logarithm is the inverse operation to exponentiation. That means the logarithm of a given number
Here
We multiply the three given logarithms in the question and have
We change the base of each logarithm to base
We use the logarithmic identity of product
We check the logarithm table and put logarithmic value in the above step to have,
So the product of logarithms is
Note: We note that if we logarithm with base 10 is called common logarithm and we have changed the base to 10 because we know about common logarithm, We note that base cannot be 1 or negative. If we do not want to use logarithm table we can use logarithmic identities inequality
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