Write the expression as a single logarithm:
2lnx+3lny-ln2-4lnz
2lnx+3lny-ln2-4lnz
ln(x^2) + ln(y^3) - ln(2) - ln(z^4)
ln(x^2y^3 / 2z^4)
ln ( a ^ n ) = n * ln ( a )
ln ( a ) + ln ( b ) = ln ( a * b )
ln ( a ) - ln ( b ) = ln ( a / b )
2 ln ( x ) + 3 ln ( y ) - ln ( 2 ) - 4 ln ( z ) =
2 ln ( x ) + 3 ln ( y ) - [ ln ( 2 ) + 4 ln ( z ) ] =
ln ( x ^ 2 * y ^ 3) - ln ( 2 z ^ 4 ) =
ln ( x ^ 2 * y ^ 3 / 2 z ^ 4 )
To write the expression as a single logarithm, we can use the properties of logarithms.
First, let's rearrange the terms in a way that groups the logarithms together:
2lnx + 3lny - ln2 - 4lnz
Next, we can use the property that states the logarithm of a product is equal to the sum of the logarithms of its factors:
ln(x^2) + ln(y^3) - ln(2) - ln(z^4)
Now, let's use the property that states the logarithm of a power is equal to the product of the exponent and the logarithm of the base:
ln(x^2 * y^3) - ln(2) - ln(z^4)
Finally, we can combine the terms by using the property that states the difference of logarithms is equal to the logarithm of the division:
ln((x^2 * y^3) / (2 * z^4))
So, the expression as a single logarithm is ln((x^2 * y^3) / (2 * z^4)).
To write the expression 2lnx + 3lny - ln2 - 4lnz as a single logarithm, we can use logarithmic rules. Specifically, we can combine the terms by applying the properties of logarithms, such as the product rule and the quotient rule.
First, let's apply the product rule, which states that ln(a) + ln(b) = ln(a * b). Using this rule, we can combine the first two terms: 2lnx + 3lny = ln(x^2) + ln(y^3) = ln(x^2 * y^3).
Next, let's apply the quotient rule, which states that ln(a) - ln(b) = ln(a / b). Using this rule, we can combine the last two terms: -ln2 - 4lnz = ln(1/2) - ln(z^4) = ln((1/2) / z^4) = ln(1/(2z^4)).
Combining the results, we have:
ln(x^2 * y^3) + ln(1/(2z^4))
Now, using the sum rule, which states that log(a) + log(b) = log(a * b), we can simplify further:
ln(x^2 * y^3) + ln(1/(2z^4)) = ln((x^2 * y^3) * (1/(2z^4))) = ln((x^2 * y^3) / (2z^4))
Therefore, the expression 2lnx + 3lny - ln2 - 4lnz can be written as a single logarithm: ln((x^2 * y^3) / (2z^4)).