Write the expression below as a single logarithm in simplest form.
4log 3−log 9
b b
The expression can be written as a single logarithm by using logarithmic identities.
Using the power rule, 4log3 can be rewritten as log3^4:
log3^4 - log9/b
Using the quotient rule, log3^4 - log9/b can be written as a single logarithm:
log((3^4)/(9/b))
Simplifying the expression inside the logarithm:
log((81)/(9/b))
Using the reciprocal rule, 9/b can be rewritten as b/9:
log((81)/(b/9))
Multiplying the numerator by the reciprocal of the denominator, we get:
log((81)*(9/b))
Multiply the numerators and the denominators:
log(729/b)
Therefore, the expression 4log3 - log(9/b) can be simplified to log(729/b).
To write the expression 4log3−log9b in a single logarithm form, we can apply the logarithm properties. Specifically, we can use the quotient rule of logarithms, which states that log(a) - log(b) is equal to log(a/b).
Using this property, we can rewrite the expression as:
4log3 - log9/b
Now, we can combine the two logarithms into a single logarithm:
log3^4 - log(9/b)
Applying the exponent property of logarithms, we get:
log81 - log(9/b)
Simplifying further, we have:
log(81/(9/b))
This can also be simplified as:
log(81 * b/9)
Thus, in simplest form, the expression 4log3−log9b is equal to log(81 * b/9).
To write the expression as a single logarithm in simplest form, we can use the logarithmic property.
The logarithmic property states that for any positive numbers a, b, and c, and any positive real number x:
1) logₐ(x) - logₐ(c) = logₐ(x/c)
2) logₐ(x) + logₐ(c) = logₐ(x * c)
3) logₐ(xⁿ) = n * logₐ(x)
Now let's simplify the given expression step by step:
4 log₃(b) - log₉(b)
Using property 3, we can rewrite log₉(b) as log₃(b²):
4 log₃(b) - log₃(b²)
Now, using property 1, we can subtract the logarithms:
log₃(b⁴) - log₃(b²)
Finally, using property 2, we can rewrite this as a single logarithm:
log₃((b⁴) / (b²))
Simplifying further, we can simplify the fraction:
log₃(b²)
So the given expression, 4 log₃(b) - log₉(b), is equivalent to log₃(b²) in simplest form.