To evaluate the given expression without using a calculator or changing its form, let's recall the definition of a logarithm.
log_b x=y ⇔ x= b^y
The logarithm y is the exponent to which b must be raised to get x. When the base of a logarithm is not stated, it means it is common and its base is 10. Therefore, we are looking for an exponent to which 10 must be raised to get 10. Recall that any number raised to the power of 1 is the number itself.
x^()darkviolet1 = x
This means that log(10) is 1.
b
Similarly as in Part A, we are given a common logarithm.
log(sqrt(10))
To evaluate the given logarithm we need to find an exponent to which 10 must be raised to get sqrt(10). Note that taking the square root of a number is the same as raising it to the power of 12.
sqrt(10) ⇔ 10^(12)
To get 10^(12), we need to raise 10 to the power of 12. Therefore, log(sqrt(10)) is 12.
c
Once again, we are given a common logarithm.
log( 0)
According to the definition of a logarithm we are looking for an exponent to which 10 has to be raised to be 0. We know that 10 raised to the power of 0 is 1. Then, 10 raised to the power of 1 is 10, and so on.
10^0=1
10^1=10
10^2=10 * 10 = 100 ...
Note that 10 raised to any power is positive. For that reason, logarithms are defined only for positive values. This means that log(0) is undefined.
d
Let's analyze the last given expression.
10^((2/3) log(27))
We are given a power of 10, where the exponent is a common logarithm multiplied by a fraction. Recall the Power of a Power Property.
a^(m n) =(a^n)^m
According to the above property, we can first calculate 10 to the power of log(27) and then raise the result to the power of 2/3. To find 10^(log(27)), we can use the Inverse Properties of Logarithms. A power and a logarithm with the same base undo each other.
b^(log_b( a)) = a
Since the base of the common logarithm is 10, 10^(log(27)) simplifies to 27.
10^(log( 27))= 27
Therefore, to evaluate the given expression we need to find 27 to the power of (2/3). Recall that the numerator of a rational exponent is the exponent of the expression, and the denominator is the index.
