Select all statements that are true.
(log_b\(A\))/(log_b\(B\))=log_b\(A-B\)
(If ) log_1.5\(8\)=x, text( then ) x**(1.5) =8. log\(500\) text( is the exponent on ) 10 text( that gives ) 500.
text
(In )log_b\(N\), text( the exponent is )N. text( If ) 2log_3\(81\)=8, text( then ) log_3\(81\) = 4
(log_b\(A\))/(log_b\(B\))=log_b\(A-B\) is not true.
If log_1.5\(8\)=x, then x**(1.5) =8 is true.
log\(500\) is the exponent on 10 that gives 500 is true.
In log_b\(N\), the exponent is N is not true.
If 2log_3\(81\)=8, then log_3\(81\) = 4 is true.
To determine which statements are true, let's analyze each one separately.
1. (log_b(A))/(log_b(B)) = log_b(A-B)
To check the validity of this statement, we need to recall the properties of logarithms. In this case, we observe that the logarithms on both sides have a common base of b. Therefore, we can rewrite the left side as log_b(A-B), which means the left side is also equal to log_b(A-B). Therefore, the statement is true.
2. If log_1.5(8) = x, then x**(1.5) = 8.
To determine the truth of this statement, we need to understand the relationship between logarithms and exponentiation. In this case, the logarithm log_1.5(8) represents the exponent to which 1.5 must be raised to obtain 8. Therefore, x is the exponent that makes 1.5^x = 8. However, the statement x**(1.5) = 8 is not valid because it suggests taking x to the power of 1.5. Hence, this statement is false.
3. log(500) is the exponent on 10 that gives 500.
To verify this statement, we need to understand the definition of logarithms. The logarithm log(500) represents the exponent to which the base (which is usually assumed to be 10 unless specified) must be raised to obtain 500. Hence, this statement is true.
4. In log_b(N), the exponent is N. If 2log_3(81) = 8, then log_3(81) = 4.
This statement relates to the understanding of logarithms. The expression 2log_3(81) means that 81 is being raised to the power of 2 first and then the logarithm base 3 is applied. If this expression equals 8, it implies that 3^8 = 81^2. To verify log_3(81) = 4, we need to check if 3^4 equals 81, which is true. Hence, this statement is also true.
In summary, the true statements from the given options are:
1. (log_b(A))/(log_b(B)) = log_b(A-B)
3. log(500) is the exponent on 10 that gives 500.
4. In log_b(N), the exponent is N. If 2log_3(81) = 8, then log_3(81) = 4.