(log\(A\))/(log\(B\)) = (ln\(A\))/(ln\(B\)) (log_b\(A\))/(log_b\(B\))=log_b\((A)/(B)\) text(In )log_b\(N\), text( the exponent is )N. text(If ) log_1.5\(8\)=x, text( then ) x**(1.5) =8. log\(500\) text( is the exponent on ) 10 text( that gives ) 500.
To understand why the equations you provided are true, we need to understand the properties of logarithms.
1. (log(A))/(log(B)) = (ln(A))/(ln(B)):
This equation is known as the Change of Base Formula for logarithms. It states that if you have a logarithm of base A divided by a logarithm of base B, it is equivalent to the natural logarithm (ln) of A divided by the natural logarithm of B. This formula allows you to convert logarithms of different bases into each other by using the natural logarithm.
2. (log_b(A))/(log_b(B)) = log_b((A)/(B)):
This equation is a property of logarithms called the Quotient Rule. It states that the logarithm of a quotient (A divided by B) with the same base is equal to the difference of the logarithms of the numerator A and the denominator B.
3. In log_b(N), the exponent is N:
This property explains what the logarithm represents. If you have a logarithm of base b with a value N, it means that b raised to the power of the logarithm equals N. In other words, N is the exponent that gives you the base b when raised to that power.
4. If log_1.5(8) = x, then x^(1.5) = 8:
This equation demonstrates the relationship between logarithms and exponents. If you have a logarithm of base 1.5 with a value x, raising x to the power of 1.5 will give you 8. This means that 1.5 raised to the power of x is equal to 8.
5. log(500) is the exponent on 10 that gives 500:
This property explains the concept of logarithms in general. If you have a logarithm of base 10 with a value of 500, it tells you that 10 raised to the power of the logarithm equals 500. In other words, the logarithm represents the exponent you need to raise the base (10 in this case) to in order to get the desired value (500).
Understanding these properties of logarithms allows you to manipulate and solve equations involving logarithmic functions.