Rewrite the expression as the sum and/or difference of logarithms, without using exponents. Simplify if possible.
Log of 7(p to the 5th root q to the 9th root) / t²
log7p^5q^9 / t^2 = log7p^5q^9-t^2 =
log7 + 5log(p) + 9log(q) - 2log(t).
To rewrite the expression as the sum and/or difference of logarithms, we can use the following properties of logarithms:
1. log(a / b) = log(a) - log(b)
2. log(xy) = log(x) + log(y)
3. log(xⁿ) = n * log(x)
Let's apply these properties to the given expression:
log(7(p^(1/5) * q^(1/9)) / t²)
Using property 2, we can split the numerator into two logarithms:
= log(7) + log(p^(1/5) * q^(1/9)) - log(t²)
Applying property 3, we can bring the exponents out front as coefficients:
= log(7) + (1/5) * log(p) + (1/9) * log(q) - 2 * log(t)
So, the expression can be rewritten as:
log(7) + (1/5) * log(p) + (1/9) * log(q) - 2 * log(t)
To rewrite the expression as the sum and/or difference of logarithms, we can use the properties of logarithms.
The first property we can use is the power rule, which states that the logarithm of a number raised to a power can be written as the product of the logarithm of the number and the exponent.
Using this property, we can rewrite the numerator:
log of 7(p to the 5th root q to the 9th root)
= log7 + log(p to the 5th root q to the 9th root)
Next, we can use the quotient rule, which states that the logarithm of a quotient can be written as the difference of the logarithm of the numerator and the logarithm of the denominator.
Using this property, we can rewrite the entire expression:
log of 7(p to the 5th root q to the 9th root) / t²
= (log7 + log(p to the 5th root q to the 9th root)) - log(t²)
Finally, if we simplify further, we can use the power rule for logarithms, which states that the logarithm of a number raised to a power can be written as the product of the power and the logarithm of the number. In this case, t² can be rewritten as 2logt.
Therefore, the expression can be simplified to:
(log7 + 5/5logp + 9/5logq) - 2logt