Questions: Find the logarithm. [ log (1,000)= ]

Find the logarithm.
[
log (1,000)=
]
Transcript text: Find the logarithm. \[ \log (1,000)= \]
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Solution

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Solution Steps

Step 1: Understanding the Logarithm

The logarithm \(\log_b a\) answers the question, "To what power must \(b\) be raised, to yield \(a\)?". In mathematical terms, if \(\log_b a = x\), then \(b^x = a\).

Step 2: Using Logarithm Properties

We can use the change of base formula to calculate the logarithm for any base. The change of base formula is \(\log_b a = \frac{\log_c a}{\log_c b}\), where \(c\) is a convenient base for calculation, typically \(e\) (natural logarithm) or 10.

Step 3: Direct Calculation for Common Bases or Using Calculators for Arbitrary Bases

Using the natural logarithm for the change of base formula, we calculate \(\log_b a = \frac{\log_e a}{\log_e b}\) = \frac{6.908}{2.303} = 3.

Final Answer:

The value of \(\log_{10} 1000\) rounded to 2 decimal places is 3.

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