How is the expression related to the expression in Part C?
A
a
x=0
B
b
x=3
C
c
x=4
D
d
x=64
Practice makes perfect
a
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
This relationship tells us that 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 a common logarithm and its base is 10. Therefore, in our case we are looking for an exponent to which 10 must be raised to get 1. Recall that any number raised to the power of is 1.
x^()darkviolet0 = 1
This means that log(1) is .
b
Similarly as in Part A, we are given a common logarithm.
log(10^3)
To evaluate the given logarithm, we need to find an exponent to which 10 must be raised to get 10^3. Note that we are already given the exponent. To get 10^3, we need to raise 10 to the power of 3. Therefore, log(10^3) is 3.
c
Again, let's analyze the given expression.
10^(log(4))We are given a power of 10, where the exponent is a common logarithm. According to 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, the given expression simplifies to 4.
10^(log( 4))= 4
d
Once again, we are given a power of 10 with a common logarithm as an exponent. However, this time the logarithm is multiplied by 3.
10^(3log(4))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(4) and then raise the result to the power of 3. In Part C, we found that 10^(log(4)) is 4. Therefore, the value of the given expression is 4 to the power of 3, which equals 64.
10^(3log(4))= 64
