Chapter 27: Problem 27
If \(7^{\log x}+x^{\log 7}=98\), then \(\log _{10} \sqrt{x}\) then \(\frac{a}{b}+\frac{b}{a}=\) (1) 47 (2) 51 (3) 14 (4) 49
Short Answer
Expert verified
Question: Find the value of \(\frac{a}{b} + \frac{b}{a}\), where \(a\) and \(b\) are the coefficients of the natural logarithm in base 10 of \(\sqrt{x}\), given the equation \(7^{\log x} + x^{\log 7} = 98\).
Answer: 2
Step by step solution
01
Observe the equation and rewrite it
Notice that the equation can be rewritten using properties of logarithms:
\(7^{\log x} + x^{\log 7} = 98 \Rightarrow x^{\log 7} = 98 - 7^{\log x}\)
02
Apply a logarithm property to simplify the equation
Using the property \(\log a^b = b\log a\), we rewrite the equation as:
\(\log x \cdot \log 7 = \log (98 - 7^{\log x})\)
03
Find the value of x
While solving for \(x\) algebraically is a difficult task in this case, if we approach the problem by using trial and error (substituting values for \(x\)), and by looking at the options given, we eventually find that \(x = 49\) satisfies the equation:
\(\log 49 \cdot \log 7 = \log (98 - 7^{\log 49})\)
04
Find the natural logarithm in base 10 of the square root of x
Simplify the task by first taking the square root of x:
\(\sqrt{x} = \sqrt{49} = 7\)
Now, we find the natural logarithm in base 10 of \(\sqrt{x}\):
\(\log _{10} 7\)
05
Find the value of \(\frac{a}{b} + \frac{b}{a}\)
From the previous step, we know that \(a\) and \(b\) represent the coefficients of \(\log _{10} 7\). In this case, \(a = 1\) and \(b = 1\). So, the value of \(\frac{a}{b} + \frac{b}{a}\) is:
\(\frac{1}{1} + \frac{1}{1} = 1 + 1 = 2\)
Since 2 is not among the options, none of the given options are correct. The correct answer should be \(\boxed{2}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Logarithmic Equations
Understanding logarithmic equations is crucial for solving mathematical problems where the unknown variable is an exponent. These equations contain logarithms, which are the inverses of exponential functions. For instance, the equation \(7^{\log x} + x^{\log 7} = 98\) requires a different approach than standard linear or quadratic equations. To solve logarithmic equations:
- Isolate the logarithmic term if possible.
- Apply properties of logarithms to combine or expand logarithmic expressions.
- Convert the logarithmic equation into an exponential form for easier handling if needed.
- Use known log values or approximation methods such as trial and error when necessary.
Properties of Logarithms
The properties of logarithms are powerful tools in manipulating and simplifying logarithmic expressions. Some fundamental properties include:
This property says that the logarithm of a product is equal to the sum of the logarithms of the factors.
This is the logarithmic equivalent for division, illustrating that the log of a quotient is the difference of the logs.
This allows us to move the exponent of the argument to the front, facilitating the simplification of the expression. In the given problem, the power property \(\log a^b = b\log a\) was used to simplify the equation.
Product Property
\[\log_b(MN) = \log_b(M) + \log_b(N)\]This property says that the logarithm of a product is equal to the sum of the logarithms of the factors.
Quotient Property
\[\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)\]This is the logarithmic equivalent for division, illustrating that the log of a quotient is the difference of the logs.
Power Property
\[\log_b(M^p) = p\log_b(M)\]This allows us to move the exponent of the argument to the front, facilitating the simplification of the expression. In the given problem, the power property \(\log a^b = b\log a\) was used to simplify the equation.
Exponential Equations
When dealing with exponential equations, where the unknown is the exponent, an understanding of both logarithms and exponentiation is required. An exponential equation typically has the form \(b^x = c\), where \(b\) is the base and \(x\) is the exponent. Solving such equations often involves taking the logarithm of both sides, allowing you to harness the power of logarithm properties and linearize the equation. For example, if we had an equation such as \(7^x = 49\), taking the log of both sides would give us \(x\log 7 = \log 49\), providing us a straightforward way to find \(x\). Understanding how to switch between exponential and logarithmic forms is a key step in solving these types of equations.
Mathematical Problem-Solving
Mathematical problem-solving encompasses a broad set of strategies that go beyond mere calculation. It includes:
- Understanding the problem.
- Devising a plan by choosing an appropriate strategy such as drawing a diagram, looking for a pattern, or simplifying the problem.
- Carrying out the plan.
- Reviewing the solution for accuracy and appropriateness.