True or False
(lna)5c = 5clna
logab^4=(logab)^4
ln 5√xy=−5(lnx+lny)
lne^3=lne^5−lne^2
lne^3=lne^5−lne^2
since e^3 = e^5/e^2
To determine whether each statement is true or false, let's break down the equations and apply the rules of logarithms:
1. (lna)5c = 5clna
This statement is false.
Explanation: According to the rules of logarithms, the exponent in front of a logarithm applies to the entire expression inside the logarithm. In this case, the exponent of 5 applies to "c" only, not to "lna." Therefore, the equation cannot be simplified to 5clna.
2. logab^4 = (logab)^4
This statement is true.
Explanation: According to the rules of logarithms, the exponent outside the entire logarithm applies to the entire expression inside the logarithm. In this case, the exponent of 4 applies to "b" only, not to "logab." Therefore, the equation can be simplified to (logab)^4.
3. ln 5√xy = −5(lnx + lny)
This statement is true.
Explanation: According to the properties of logarithms, when taking the natural logarithm of a product or a quotient, it can be split into the sum or difference of the natural logarithms of the individual terms. In this case, the natural logarithm of 5√xy can be split into the sum of ln 5, ln √x, and ln y. Then, by applying the rules of logarithms, ln √x can be written as ln x^(1/2), which simplifies to (1/2)lnx. Finally, we multiply the entire equation by -5 to obtain −5(lnx + lny) on the right side.
4. lne^3 = lne^5 − lne^2
This statement is true.
Explanation: The natural logarithm of e (ln e) is equal to 1. Therefore, lne^3 simplifies to 3 and lne^2 simplifies to 2. Thus, the equation becomes 3 = 5 - 2, which is 3 = 3. Since the equation is true, the statement is true.
Remember to always apply the rules of logarithms correctly to determine the accuracy of equations.