Evaluate the following: log_10(24) + 2 * log_10(15) – 4 * log_10(125)
The Correct Answer and Explanation is :
The evaluated result of the given logarithmic expression:
[
\log_{10}(24) + 2 \cdot \log_{10}(15) – 4 \cdot \log_{10}(125)
]
is -4.655 (approximately).
Explanation:
We simplify the expression step by step using logarithmic properties.
Step 1: Expanding Logarithmic Terms
Using the logarithmic identity:
[
\log(a^b) = b \log(a)
]
we rewrite:
[
2 \log_{10}(15) = \log_{10}(15^2) = \log_{10}(225)
]
[
4 \log_{10}(125) = \log_{10}(125^4) = \log_{10}(244140625)
]
Thus, the given expression transforms into:
[
\log_{10}(24) + \log_{10}(225) – \log_{10}(244140625)
]
Step 2: Applying Logarithmic Properties
Using the identity:
[
\log(a) + \log(b) = \log(a \cdot b)
]
we combine:
[
\log_{10}(24 \times 225) = \log_{10}(5400)
]
Then applying:
[
\log(a) – \log(b) = \log\left(\frac{a}{b}\right)
]
we simplify:
[
\log_{10} \left(\frac{5400}{244140625}\right)
]
Step 3: Calculating the Logarithm
Computing:
[
\frac{5400}{244140625} \approx 2.211 \times 10^{-5}
]
Taking the base-10 logarithm:
[
\log_{10}(2.211 \times 10^{-5}) = \log_{10}(2.211) + \log_{10}(10^{-5})
]
Approximating:
[
\log_{10}(2.211) \approx 0.345
]
[
\log_{10}(10^{-5}) = -5
]
So,
[
0.345 – 5 = -4.655
]
Thus, the final result is -4.655.
Summary:
The logarithmic expression simplifies to approximately -4.655, and the graph above represents the logarithmic function, highlighting key values.