Use properties of logarithms to find the exact value of this expression. Do not use a calculator.
6log6 8 - log6 7
To find the exact value of the expression 6log6(8) - log6(7), we can use the properties of logarithms.
1. Start by applying the power rule, which states that log base b of (a^n) is equal to n times log base b of a.
So, we have: 6log6(8) - log6(7) = log6(8^6) - log6(7)
2. Simplify the expression inside the parentheses. 8^6 can be calculated without a calculator as follows:
8^6 = (2^3)^6 = 2^(3*6) = 2^18
3. Now, our expression becomes: log6(2^18) - log6(7)
4. Apply the product rule of logarithms, which states that log base b of (a * c) is equal to the log base b of a plus the log base b of c.
Therefore, we can rewrite our expression as log6(2^18 / 7).
5. Since the base of the logarithm is 6, we want to express 2^18 / 7 as a power of 6.
6. Simplify 2^18 / 7: 2^18 / 7 = (2^18) / 7 = (2^18) / (2^3) = 2^(18-3) = 2^15
7. Substitute back into the expression: log6(2^15)
8. Finally, we can apply the definition of logarithms, which states that log base b of a is the exponent to which b must be raised to obtain a.
Therefore, log6(2^15) = 15.
So, the exact value of the expression 6log6(8) - log6(7) is 15.