Express each of the following as a single logarithmic expression. All exponents must be left in radical form:
1.) 2logx-3logy
2.) logx-logy+logz
3.) 1/3log5x+2/3log6x
4.) 2/3(logbase2 x - logbase2 y)
I don't understand these problems at all. Can someone please solve and explain them? Thanks
To combine the two log terms, first, the terms must have the same base (the number subscript of log). In this case, if there is no number subscript of log, the base is equal to 10. The two terms have the same base, 10.
Now, recall that to combine log terms of the same base,
*if addition: multiply the terms inside the log. for example
log 3 + log 5 = log (3*5) = log 15
*if subtraction: divide the terms inside the log. for example,
log 3 - log 5 = log (3/5)
*also, if the log is multiplied by a number (outside), we can rewrite it as exponent of the term inside the log. for example,
2log 3 = log (3^2) = log 9
#1.
applying these rules,
2log x - 3log y
we first make the number outside as an exponent to the term inside the log:
log x^2 - log y^3
since subtraction, we divide the terms inside the log:
log (x^2 / y^3)
#2.
log x - log y + log z
log (x/y) + log z
log (xz/y)
#3.
1/3log 5x + 2/3 log 6x
log (5x)^(1/3) + log (6x)^(2/3)
log [(5x)^(1/3) * (6x)^(2/3)]
**recall that to multiply terms with the same base, we add their exponents. for example,
a^(1/3) * a^(2/3) = a^(1/3 + 2/3) = a^(3/3) = a
**also, we can rewrite fraction exponents as, for instance,
2^(2/3) = cuberoot(2^2) = cuberoot (4)
going back to the problem,
log [(5x)^(1/3) * (6x)^(2/3)]
log [5^(1/3) * 6^(2/3) * x^(1/3 + 2/3)]
log [cuberoot(5*6^2) * x^(3/3)]
log [cuberoot(5*36)x]
log [cuberoot(180)x]
#4.
2/3(logbase2 x - logbase2 y)
note that the base now is 2, but that's not a problem since both terms have the same base.
2/3[ logbase2 (x/y) ]
logbase2 (x/y)^(2/3) or
logbase2 [cuberoot (x^2)/(y^2)]
hope this helps~ :)
Sure! Let's go through each problem step by step.
1.) 2logx - 3logy
To express this as a single logarithmic expression, we need to combine the terms.
First, let's apply the power rule of logarithms, which states that log a^b = b*log a:
2logx - 3logy = logx^2 - logy^3
Now, we can use the quotient rule of logarithms, which states that log a - log b = log (a/b):
logx^2 - logy^3 = log(x^2/y^3)
So, the final expression is log(x^2/y^3).
2.) logx - logy + logz
Similarly, we can combine the terms in this expression.
Using the quotient rule, we have:
logx - logy + logz = log(x/z) - logy
So, the expression can be simplified to log(x/z) - logy.
3.) (1/3)log5x + (2/3)log6x
In this case, we have coefficients in front of the logarithms. We can use the power rule to move the coefficients as exponents inside the logarithms:
(1/3)log5x + (2/3)log6x = log((5x)^(1/3)) + log((6x)^(2/3))
Next, we can use the product rule, which states that log a + log b = log(ab), to combine the two logarithms:
log((5x)^(1/3)) + log((6x)^(2/3)) = log(((5x)^(1/3)) * ((6x)^(2/3)))
Simplifying the expression inside the logarithm gives us:
log((5^(1/3) * x^(1/3)) * (6^(2/3) * x^(2/3)))
= log((5^(1/3) * 6^(2/3)) * (x^(1/3) * x^(2/3)))
= log((5^(1/3) * 6^(2/3)) * x)
Therefore, the final expression is log((5^(1/3) * 6^(2/3)) * x).
4.) (2/3)(log base 2 x - log base 2 y)
In this problem, we have a logarithm with a base other than 10. To express it as a single logarithmic expression, we can use the quotient rule:
(2/3)(log base 2 x - log base 2 y) = (2/3)log base 2 (x/y)
So, the final expression is (2/3)log base 2 (x/y).
I hope this helps! Let me know if you have any further questions.