(b) log10(0.01) = x
x=
The correct answer and explanation is :
Solution:
We need to determine the value of ( x ) in the equation:
[
\log_{10}(0.01) = x
]
First, recall the definition of a logarithm:
[
\log_b(A) = C \quad \text{means} \quad b^C = A
]
In this case, ( b = 10 ), ( A = 0.01 ), and we need to find ( x ) such that:
[
10^x = 0.01
]
Now, express ( 0.01 ) as a power of 10:
[
0.01 = 10^{-2}
]
Thus, we can rewrite the equation as:
[
10^x = 10^{-2}
]
Since the bases are the same, we equate the exponents:
[
x = -2
]
Final Answer:
[
\log_{10}(0.01) = -2
]
Explanation (300 Words):
Logarithms are mathematical tools used to express exponents. The logarithmic function ( \log_b(A) = C ) is the inverse of an exponential function and tells us what exponent ( C ) we need to raise ( b ) to in order to get ( A ).
In this problem, we are given the base-10 logarithm of 0.01 and asked to determine its value. The logarithm of a number less than 1 is always negative because raising 10 to a positive exponent always results in a number greater than 1.
To solve ( \log_{10}(0.01) ), we rewrite 0.01 as a power of 10:
[
0.01 = 1/100 = 10^{-2}
]
Using the logarithm rule:
[
\log_{10}(10^C) = C
]
we substitute ( 10^{-2} ) in place of ( 0.01 ):
[
\log_{10}(10^{-2}) = -2
]
This means that ( 10^{-2} = 0.01 ), confirming our result.
Why is the Answer Negative?
The logarithm of a number less than 1 is negative because the exponent needed to produce a fraction (like 0.01) is negative. This happens because dividing by 10 moves the decimal point to the left, making the number smaller.
Thus, the final answer is ( x = -2 ).