Question

Find the value of \(x\) and \(y\) respectively for \(\log _{10}\left(x^{2} y^{3}\right)=7 \quad\) and \(\quad \log _{10}(x / y)=1:\) (a) \(x=10, y=100\) (b) \(x=100, y=10\) (c) \(x=10, y=20\) (d) none of these

Short answer

Solution: Given logarithmic equations are: 1. \(\log_{10}(x^2y^3) = 7\) 2. \(\log_{10}(x/y) = 1\) After solving the given equations, we find that the values of x and y are: x = 100 and y = 10 Thus, the correct answer is x=100 and y=10.

Step-by-step solution

Convert logarithmic equations to exponential equations

First, we need to convert the given logarithmic equations into exponential equations using the fact that \(\log_b{a^m}=m\log_b{a}\). For Equation 1: - Apply the logarithmic rule \(\log{a^m} = m\log{a}\): \(\log_{10}(x^2y^3)=2\log_{10}x + 3\log_{10}y = 7\) For Equation 2: - Apply the logarithmic rule \(\log{a/b} = \log{a} - \log{b}\): \(\log_{10}(x / y) = \log_{10}x - \log_{10}y = 1\) Now the equations are: 1. \(2\log_{10}x + 3\log_{10}y = 7\) 2. \(\log_{10}x - \log_{10}y = 1\)

Solve the system of equations

We will solve the system of equations using the substitution or elimination method. Here, we will use the substitution method. From Equation 2: \(\log_{10}x = 1+\log_{10}y\) Substitute this expression for \(\log_{10}x\) in Equation 1: \(2(1 + \log_{10}y) + 3\log_{10}y = 7\) Simplify the equation: \(2 + 5\log_{10}y = 7\) Now isolate \(\log_{10}y\): \(\log_{10}y = 1\)

Find the value of \(y\)

Now, using the exponential form for logarithms, we can find the value of \(y\): \(10^1 = y \Rightarrow y = 10\)

Find the value of \(x\)

Now that we have the value of \(y\), we can substitute it back into either equation to find the value of \(x\). We will use Equation 2: \(\log_{10}x - \log_{10}10 = 1\) Add \(\log_{10}10\) to both sides: \(\log_{10}x = 2\) Using the exponential form for logarithms: \(10^2 = x \Rightarrow x = 100\)

Check the given options and choose the correct one

We have found the values of \(x\) and \(y\): \(x = 100\) and \(y = 10\) Comparing these values to the given options: (a) \(x=10, y=100\) (b) \(x=100, y=10\) (correct) (c) \(x=10, y=20\) (d) none of these Therefore, the correct answer is \(\boxed{\text{(b)}\, x=100,\, y=10}\).

Key concepts

Exponential Equations

Exponential equations are mathematical expressions where variables appear as exponents. For example, in the equation \( 10^x = 100 \), \( x \) is the exponent. To solve these types of equations, we often use the properties of exponents and logarithms.

Solving an exponential equation often involves converting to a logarithmic form because logarithms are the inverses of exponents. This conversion allows us to isolate the variable and find its value. For instance, if we have \( 10^x = 100 \), we can take the logarithm of both sides to obtain \( x\text{log}(10) = \text{log}(100) \), and since \( \text{log}(10) = 1 \), we find \( x = 2 \).

Understanding the relationship between exponents and logarithms is crucial for solving exponential equations effectively. They are fundamental in various fields such as compound interest calculations in finance, population growth models in biology, and the decay of radioactive materials in physics.

Solving Logarithmic Problems

Logarithmic problems involve equations where the unknown variable is part of the logarithm. Solving these requires a good grasp of logarithmic properties, such as the product rule \( \text{log}_b(a \times c) = \text{log}_b(a) + \text{log}_b(c) \), the quotient rule \( \text{log}_b(a / c) = \text{log}_b(a) - \text{log}_b(c) \), and the power rule \( \text{log}_b(a^m) = m\text{log}_b(a) \).

These properties help us manipulate the equation into a more solvable form. For example, to solve \( \text{log}_b(x^2y^3) = 7 \), we use the property of logarithms to rewrite it as a sum: \( 2\text{log}_b(x) + 3\text{log}_b(y) = 7 \). We can then proceed to isolate each variable and solve for their values using the inverse operations—exponentiation—and the known properties of logarithms.

System of Equations

A system of equations is a set of two or more equations with multiple variables. The goal is to find the values of the variables that satisfy all the equations simultaneously. There are different methods to solve systems of equations, such as graphing, substitution, elimination, and using matrices.

In the context of logarithmic equations, the substitution method often comes in handy. For example, if we have two equations \( 2\text{log}_{10}x + 3\text{log}_{10}y = 7 \) and \( \text{log}_{10}x - \text{log}_{10}y = 1 \), we can solve for one variable in terms of the other using one equation and then substitute this expression into the other equation. This allows us to solve for one of the variables and then backtrack to find the other.

The ability to solve systems of equations is not only crucial in algebra but also in various applied fields such as economics, physics, and engineering, where multiple conditions must be met simultaneously.