use the properties of logarithms to find the exact value of the expression
loglower 3^18 - log lower3^6
**how do I make the three lower in the log equation?
To write the subscript (index) of a logarithmic expression, you can use the underscore (_) symbol followed by the number. Here's the modified equation with the subscripts included:
log₃(3^18) - log₃(3^6)
Now, let's simplify the expression using the properties of logarithms.
1. Power Rule: logₐ(b^c) = c * logₐ(b)
Using the power rule, we can simplify the first term:
log₃(3^18) = 18 * log₃(3)
2. Division Rule: logₐ(b/c) = logₐ(b) - logₐ(c)
Next, let's simplify the second term using the division rule:
log₃(3^6) = log₃(3^6) = 6 * log₃(3)
Now that we've simplified both terms, let's substitute them back into the original expression:
18 * log₃(3) - 6 * log₃(3)
Since both terms have the same base (3), we can combine them using the subtraction rule:
18 * log₃(3) - 6 * log₃(3) = log₃(3^(18-6))
Simplifying further:
= log₃(3^12)
= log₃(531441)
Therefore, the exact value of the expression log₃(3^18) - log₃(3^6) is log₃(531441).