If log 10 6 = 0.7782 and log 10 8 = 0.9031, then what is the value of log 10 8000 + log 10 600?
6.6813
This solution explains how to find the value of log 10 8000 + log 10 600 using the given logarithm values and properties.
Logarithm Properties Recap
To solve this problem, we need to understand a few key properties of logarithms:
- Product Rule: The logarithm of a product is the sum of the logarithms. Mathematically, this is expressed as: \( \log_b (x \times y) = \log_b x + \log_b y \)
- Power Rule: The logarithm of a number raised to a power is the power times the logarithm of the number. Mathematically: \( \log_b (x^n) = n \times \log_b x \)
- Base Rule: The logarithm of the base itself is always 1. For base 10: \( \log_{10} 10 = 1 \)
- Logarithm of Powers of 10: Combining the power rule and the base rule, we get: \( \log_{10} (10^n) = n \times \log_{10} 10 = n \)
We are given:
- \( \log_{10} 6 = 0.7782 \)
- \( \log_{10} 8 = 0.9031 \)
Evaluating log 10 8000
We need to express 8000 in a way that uses the given logarithm values or powers of 10.
We can write 8000 as \( 8 \times 1000 \). Therefore:
\( \log_{10} 8000 = \log_{10} (8 \times 1000) \)Using the product rule for logarithms:
\( \log_{10} 8000 = \log_{10} 8 + \log_{10} 1000 \)Since \( 1000 = 10^3 \), we have:
\( \log_{10} 1000 = \log_{10} 10^3 \)Using the logarithm of powers of 10 rule (\( \log_{10} 10^n = n \)):
\( \log_{10} 10^3 = 3 \)Substituting this back:
\( \log_{10} 8000 = \log_{10} 8 + 3 \)Now, substitute the given value \( \log_{10} 8 = 0.9031 \):
\( \log_{10} 8000 = 0.9031 + 3 = 3.9031 \)Evaluating log 10 600
Similarly, we express 600 using the given values or powers of 10.
We can write 600 as \( 6 \times 100 \). Therefore:
\( \log_{10} 600 = \log_{10} (6 \times 100) \)Using the product rule:
\( \log_{10} 600 = \log_{10} 6 + \log_{10} 100 \)Since \( 100 = 10^2 \):
\( \log_{10} 100 = \log_{10} 10^2 \)Using the logarithm of powers of 10 rule:
\( \log_{10} 10^2 = 2 \)Substituting this back:
\( \log_{10} 600 = \log_{10} 6 + 2 \)Now, substitute the given value \( \log_{10} 6 = 0.7782 \):
\( \log_{10} 600 = 0.7782 + 2 = 2.7782 \)Calculating the Sum log 10 8000 + log 10 600
We need to find the sum of the two values we calculated:
\( \log_{10} 8000 + \log_{10} 600 = 3.9031 + 2.7782 \)Adding these values:
\( 3.9031 + 2.7782 = 6.6813 \)Alternative Calculation Method
We can also solve this by combining the terms first:
\( \log_{10} 8000 + \log_{10} 600 \)Substitute the expanded forms from the previous steps:
\( = (\log_{10} 8 + 3) + (\log_{10} 6 + 2) \)Rearrange the terms:
\( = \log_{10} 8 + \log_{10} 6 + 3 + 2 \) \( = \log_{10} 8 + \log_{10} 6 + 5 \)Now substitute the given values \( \log_{10} 8 = 0.9031 \) and \( \log_{10} 6 = 0.7782 \):
\( = 0.9031 + 0.7782 + 5 \)Perform the addition:
\( = 1.6813 + 5 \) \( = 6.6813 \)Final Value Determination
Both methods confirm that the value of the expression log 10 8000 + log 10 600 is 6.6813.
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