Express the following as a logarithm and simplify.
1. log(3x^2 + 11x - 20) - log(3x - 4)
2. 1/6 log3 27 + log3 18 - log3 2
3. log5 cube root of 625
4. 2loga x^3 + 3loga y + 4loga z - 3logz w
since 3x^2+11x-20 = (3x-4)(x+5) you have
log(3x-4)+log(x+5)-log(3x-4) = log(x+5)
log3(27) = 3
log(18) = log(9) + log(2)
So, you have
1/6 (3) + 2 + log3(2) - log3(2) = 5/2
625 = 5^4, so you have
log5(5^(4/3)) = 4/3
surely you meant loga w. If so, you have
log_a(x^6 y^3 z^4 w^3)
To express and simplify logarithmic expressions, we can apply certain rules and properties of logarithms. Let's go through each expression.
1. log(3x^2 + 11x - 20) - log(3x - 4):
To simplify this expression, we can use the quotient property of logarithms, which states that log(a) - log(b) is equivalent to log(a/b).
Thus, we can rewrite the expression as:
log[(3x^2 + 11x - 20)/(3x - 4)]
This is the simplified form of the expression.
2. 1/6 log3 27 + log3 18 - log3 2:
In this expression, we have three terms. We can use the power rule of logarithms, which states that log(base b) a^c is equal to c*log(base b) a.
Let's simplify each term individually:
1/6 log3 27 = 1/6 * 3 = 1/2 (since 3^3 = 27)
log3 18 = log3 (2 * 3^2) = log3 2 + log3 3^2 = log3 2 + 2
log3 2 remains as it is.
Combining these terms, we get:
1/2 + (log3 2 + 2) - log3 2
Since log3 2 and -log3 2 cancel each other out, we are left with:
1/2 + 2 = 2.5
Thus, the simplified form of the expression is 2.5.
3. log5 cube root of 625:
To express this expression as a logarithm, we can use the power rule of logarithms. The cube root of 625 can be written as 5^(2/3), since (5^(2/3))^3 = 625.
Therefore, the expression can be written as:
log5 (5^(2/3))
Since the base of the logarithm (5) and the base of the exponent (5) are the same, the expression simplifies to:
2/3
Thus, the simplified form of the expression is 2/3.
4. 2loga x^3 + 3loga y + 4loga z - 3logz w:
To simplify this expression, we can use the power rule of logarithms. Recall that log(base b) a^c is equal to c*log(base b) a.
Let's simplify each term individually:
2loga x^3 = loga (x^3)^2 = loga x^6
3loga y remains as it is
4loga z remains as it is
-3logz w remains as it is
Combining these terms, we get:
loga x^6 + 3loga y + 4loga z - 3logz w
This is the simplified form of the expression.