simplify each expression by using properties of logarithms or definitions. check your results using the change of base formula
a.) log3 27=
b.) log5 (1/5)=
c.) if f(x)= ln (x), find f(e^x)
(a) 3log (3^3) = 3
(b) 5log (5^(-1)) = -1
(c) f(e^x) = ln(e^x) = x
a.) To simplify the expression log3 27, we can use the property of logarithms which states that log base b of a to the power of c is equal to c times log base b of a.
In this case, we have log3 27. We know that 27 is equal to 3 to the power of 3 (27 = 3^3). So we can rewrite the expression as log3 (3^3).
According to the property mentioned above, we can rewrite it as 3 times log3 3. Since the logarithm of the base to itself is equal to 1, log3 3 equals 1. Therefore, the expression simplifies to 3 times 1, which is 3.
To check our result using the change of base formula, we can rewrite the expression as log3 27 / log3 10, where 10 is the base we choose for our new logarithm. Simplifying further, we have log10 27 / log10 3. Evaluating this using a calculator, we get 1.431.
b.) To simplify the expression log5 (1/5), we can use a property of logarithms which says that log base b of 1/a is equal to -log base b of a.
In this case, we have log5 (1/5). Using the property mentioned earlier, we can rewrite it as -log5 5.
Since the logarithm of the base to itself is equal to 1, -log5 5 is equal to -1.
To check our result using the change of base formula, we can rewrite the expression as log5 (1/5) / log5 10, where 10 is the base we choose for our new logarithm. Simplifying further, we have log10 (1/5) / log10 5. Evaluating this using a calculator, we get -1.
c.) To find f(e^x) given f(x) = ln(x), we substitute e^x in place of x in the equation f(x) = ln(x).
So, f(e^x) = ln(e^x).
By the definition of the natural logarithm, ln(e^x) simplifies to x.
Therefore, f(e^x) = x.
No further steps are needed to simplify this expression.