Logarithm Simplification Strategy
The goal is to express \(\log_{10}(384)\) in terms of \(p\), \(q\), and \(r\). We first find the prime factorization of each number involved.
- \(80 = 10 \times 8 = (2 \times 5) \times 2^3 = 2^4 \times 5\)
- \(45 = 9 \times 5 = 3^2 \times 5\)
- \(216 = 6^3 = (2 \times 3)^3 = 2^3 \times 3^3\)
- \(384 = 128 \times 3 = 2^7 \times 3\)
Expressing p, q, and r
Using the properties of logarithms (\(\log(ab) = \log(a) + \log(b)\) and \(\log(a^n) = n\log(a)\)), we rewrite \(p\), \(q\), and \(r\):
- \(p = \log_{10}(80) = \log_{10}(2^4 \times 5) = 4\log_{10}(2) + \log_{10}(5)\)
- \(q = \log_{10}(45) = \log_{10}(3^2 \times 5) = 2\log_{10}(3) + \log_{10}(5)\)
- \(r = \log_{10}(216) = \log_{10}(2^3 \times 3^3) = 3\log_{10}(2) + 3\log_{10}(3)\)
Target Logarithm Expression
The expression we need to find is:
\(\log_{10}(384) = \log_{10}(2^7 \times 3) = 7\log_{10}(2) + \log_{10}(3)\)
Combining p, q, and r
Let's test the combination \(p - q + r\) using the expressions derived above:
\(p - q + r = (4\log_{10}(2) + \log_{10}(5)) - (2\log_{10}(3) + \log_{10}(5)) + (3\log_{10}(2) + 3\log_{10}(3))\)
Combine like terms:
\(= (4\log_{10}(2) + 3\log_{10}(2)) + (-2\log_{10}(3) + 3\log_{10}(3)) + (\log_{10}(5) - \log_{10}(5))\)
\(= 7\log_{10}(2) + 1\log_{10}(3) + 0\)
\(= 7\log_{10}(2) + \log_{10}(3)\)
This matches the target expression for \(\log_{10}(384)\).
Conclusion
Therefore, \(\log_{10}(384)\) is equal to \(p - q + r\).
Similar Questions
If \(a^b = b^a\), then what is
\(\frac{a \times (\frac{a}{b}) ^{ \frac{a}{b} } }{ (a)^{ \frac{a}{b} } }\)
equal to?
- If \(\alpha\) and \(\beta\) are the roots of the equation \(\log_{10} [998+\sqrt{x^2-18x+76}] = 3\) then what is \((\alpha - \beta)^2\) equal to?
If \(x^4 + y^4 = 14x^2 y^2\), then consider the following :
I. \(\log_{10}(x^2 + y^2)\) = \(\log_{10} x + \log_{10} y + 2\log_{10} 2\)
II. \(\log_{10}(x^2-y^2)=\log_{10} x + \log_{10} y\) +\(\log_{10}2+0.5\log_{10} 3\)
Which of the above is/are correct?
- If \(\log_{10}2 = 0.301\) and \(\log_{10} 3 = 0.477\), then what is the number of digits in the expansion of \(60^{60}\) ?
If log10(100001 - 4x)/(5 - x) = 1, then what is x equal to?
What is √17 - 4√15 + √8 - 2√15 equal to?
A sum of money at the rate of \(5\%\) per annum compounded annually becomes \(n\) times in \(100\) years. What is the value of \(n\)? (Given \(\log_{10}2=0.301\), \(\log_{10}3=0.477\) and \(\log_{10}7=0.845\))
What is the number of zeros immediately after the decimal point in \((0.5)^{1000}\)? (Given that \(\log_{10}2 = 0.30103\))
Consider the following statements in respect of common logarithms :
I. The logarithm of a number greater than 100 but less than 1000 lies between 2 and 3.
II. The logarithm of a positive number less than unity is negative.
Which of the statements given above is/are correct?
Important Questions from Logarithms
Given that $132^{0.14} = x$, $132^{0.26} = y$ and $x^z = y^2$, then the value of z is close to:
- If \(p + q = 15\), then what is \(q-p\) equal to?
- If \(p + q = 66\), then which one of the following is correct?
For \(x \ge y > 1\), let \(\log_x\left(\frac{x}{y}\right) + \log_y\left(\frac{y}{x}\right) = k\), then the value of \(k\) can never be equal to
If \(\log_ba = p\), \(\log_dc = 2p\) and \(\log_fe = 3p\), then what is \((ace)^{\frac{1}{p}}\) equal to ?
