(a) log10(1000) = x
x=
The correct answer and explanation is :
Solution:
[
\log_{10}(1000) = x
]
Since logarithms answer the question: “10 raised to what power gives 1000?” we rewrite it as:
[
10^x = 1000
]
We know that:
[
1000 = 10^3
]
So, we substitute:
[
10^x = 10^3
]
Since the bases are the same, we can equate the exponents:
[
x = 3
]
Thus, the correct answer is:
[
x = 3
]
Explanation (300 words):
Logarithms are the inverse of exponentiation, meaning they help us determine the exponent needed to reach a certain value when using a specific base. In this case, we are working with base 10, known as the common logarithm. The equation given is:
[
\log_{10}(1000) = x
]
To solve for ( x ), we need to find the exponent to which 10 must be raised to produce 1000. Since exponentiation is the reverse of logarithms, we rewrite it as:
[
10^x = 1000
]
Next, we express 1000 as a power of 10:
[
1000 = 10^3
]
By comparing exponents on both sides:
[
10^x = 10^3
]
Since the bases are identical, the exponents must be equal:
[
x = 3
]
Thus, the final answer is:
[
\log_{10}(1000) = 3
]
This result aligns with the general logarithmic property:
[
\log_b(b^n) = n
]
For base 10:
[
\log_{10}(10^n) = n
]
In everyday applications, logarithms are widely used in fields such as engineering, physics, and computer science. For example, they are essential in measuring the pH scale in chemistry, the Richter scale for earthquakes, and decibels in sound measurement.
Understanding logarithms simplifies complex calculations and helps analyze exponential growth or decay in real-world scenarios like population growth, radioactive decay, and financial interest calculations.