There are n zeros appearing immediately after the decimal point in the value of (0.2) 25 . It is given that the value of log 10 2 = 0.30103. The value of n is
17
Understanding the Number of Zeros After Decimal Point
The question asks for the number of zeros that appear immediately after the decimal point in the value of \((0.2)^{25}\). This type of problem is typically solved using logarithms, specifically the characteristic of the common logarithm (base 10) of the number.
The number of zeros immediately after the decimal point in a number \(N < 1\) is related to the characteristic of its common logarithm, \(\log_{10}(N)\). If the characteristic of \(\log_{10}(N)\) is \(-c\), where \(c\) is a positive integer, then the number of zeros immediately after the decimal point is \(c-1\).
Calculating the Logarithm of \( (0.2)^{25} \)
Let's calculate the logarithm of the given number, \((0.2)^{25}\). We are given that \(\log_{10} 2 = 0.30103\).
We need to find \(\log_{10} ((0.2)^{25})\). Using the property of logarithms, \(\log a^b = b \log a\):
\[ \log_{10} ((0.2)^{25}) = 25 \times \log_{10} (0.2) \]Using another property of logarithms, \(\log (a/b) = \log a - \log b\):
\[ \log_{10} (0.2) = \log_{10} \left( \frac{2}{10} \right) = \log_{10} 2 - \log_{10} 10 \]We know that \(\log_{10} 10 = 1\) and we are given \(\log_{10} 2 = 0.30103\). Substituting these values:
\[ \log_{10} (0.2) = 0.30103 - 1 = -0.69897 \]Now substitute this back into the expression for \(\log_{10} ((0.2)^{25})\):
\[ \log_{10} ((0.2)^{25}) = 25 \times (-0.69897) \]
Let's perform the multiplication:
\[ 25 \times 0.69897 = 17.47425 \]
So, \(\log_{10} ((0.2)^{25}) = -17.47425\).
Determining the Characteristic and Number of Zeros
The value of \(\log_{10} ((0.2)^{25})\) is \(-17.47425\). We need to express this in the form Characteristic + Mantissa, where the mantissa is non-negative.
\[ -17.47425 = -18 + 0.52575 \]
In this expression, the characteristic is \(-18\) and the mantissa is \(0.52575\).
The characteristic of the logarithm is \(-18\). As discussed earlier, if the characteristic is \(-c\), the number of zeros immediately after the decimal point is \(c-1\).
Here, the characteristic is \(-18\), so \(c=18\). The number of zeros \(n\) is:
\[ n = c - 1 = 18 - 1 = 17 \]
Therefore, there are 17 zeros appearing immediately after the decimal point in the value of \((0.2)^{25}\).
Step-by-Step Calculation Summary
- Find \(\log_{10} (0.2)\).
- Calculate \(\log_{10} ((0.2)^{25})\) by multiplying by 25.
- Determine the characteristic of the resulting logarithm.
- Use the characteristic to find the number of zeros immediately after the decimal point.
| Step | Calculation | Result | Notes |
|---|---|---|---|
| 1 | \(\log_{10} (0.2) = \log_{10} 2 - \log_{10} 10\) | \(0.30103 - 1\) | Using \(\log(a/b)\) property |
| 2 | \(-0.69897\) | Value of \(\log_{10} (0.2)\) | |
| 3 | \(\log_{10} ((0.2)^{25}) = 25 \times \log_{10} (0.2)\) | \(25 \times (-0.69897)\) | Using \(\log a^b\) property |
| 4 | \(-17.47425\) | Value of \(\log_{10} ((0.2)^{25})\) | |
| 5 | Characteristic of \(-17.47425\) | \(-18\) | Integer part when mantissa is non-negative |
| 6 | Number of zeros \(n = |\text{Characteristic}| - 1\) | \(18 - 1\) | Applying the rule for zeros after decimal |
| 7 | \(17\) | Final number of zeros |
The value of \(n\), the number of zeros appearing immediately after the decimal point, is 17.
Revision Table: Logarithm Characteristics and Zeros
| Number \(N\) | \(\log_{10} N\) | Characteristic | Number of Zeros After Decimal |
|---|---|---|---|
| 0.5 | \(\approx -0.30103 = -1 + 0.69897\) | -1 | \(1 - 1 = 0\) (e.g., 0.5) |
| 0.05 | \(\approx -1.30103 = -2 + 0.69897\) | -2 | \(2 - 1 = 1\) (e.g., 0.05) |
| 0.005 | \(\approx -2.30103 = -3 + 0.69897\) | -3 | \(3 - 1 = 2\) (e.g., 0.005) |
| \(N < 1\) with characteristic \(-c\) | \(-c + \text{mantissa}\) (mantissa \(\ge 0\)) | \(-c\) | \(c - 1\) |
Additional Information: Logarithm Properties
Logarithms are powerful tools for simplifying calculations involving multiplication, division, and exponents. The common logarithm (base 10) is particularly useful for determining the number of digits in a large number or the number of zeros after the decimal point in a small number.
- Definition: \(\log_b a = c\) means \(b^c = a\). For common logs, \(\log_{10} a = c\) means \(10^c = a\).
- Product Rule: \(\log_b (xy) = \log_b x + \log_b y\)
- Quotient Rule: \(\log_b (x/y) = \log_b x - \log_b y\)
- Power Rule: \(\log_b x^p = p \log_b x\)
- Change of Base: \(\log_b a = \frac{\log_c a}{\log_c b}\)
The characteristic of \(\log_{10} N\) tells us about the magnitude of the number \(N\).
- If \(N \ge 1\), the characteristic is a non-negative integer equal to (number of digits before the decimal point - 1).
- If \(0 < N < 1\), the characteristic is a negative integer. If the characteristic is \(-c\), then the first non-zero digit appears in the \(c^{\text{th}}\) decimal place, and there are \((c-1)\) zeros immediately after the decimal point.
In our case, \(\log_{10} ((0.2)^{25}) = -17.47425\). The characteristic is \(-18\). This means the first non-zero digit in \((0.2)^{25}\) appears in the 18th decimal place, and there are \(18-1 = 17\) zeros immediately after the decimal point.
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