Question

If \(\log \left(\frac{a+b}{6}\right)=\frac{1}{2}(\log a+\log b)\), then \(\frac{a}{b}+\frac{b}{a}=\) (1) 30 (2) 31 (3) 32 (4) 34

Short answer

Question: Given the equation \(\log \left(\frac{a+b}{6}\right)=\frac{1}{2}(\log a+\log b)\), find the value of \(\frac{a}{b}+\frac{b}{a}\). Answer: (4) 34.

Step-by-step solution

Use logarithm properties to simplify the equation.

We can simplify the given equation using the logarithm properties \(\log x^n=n\log x\) and \(\log (x*y) = \log x + \log y\). The given equation is: \(\log \left(\frac{a+b}{6}\right)= \frac{1}{2}(\log a+\log b)\) Use properties of logarithms to simplify the equation: \(\log ((a+b)/6)=\log(a^{\frac{1}{2}}*b^{\frac{1}{2}})\)

Remove logarithms.

Since the logarithms on both sides are equal, we can conclude that: \((a+b)/6=a^{\frac{1}{2}}*b^{\frac{1}{2}}\)

Get rid of the square roots.

Multiply both sides by 6: \(a + b = 6(a^{\frac{1}{2}}*b^{\frac{1}{2}})\) Square both sides to get rid of the square roots: \((a + b)^2 = 36ab\) Expand the square: \(a^2 + 2ab + b^2 = 36ab\)

Manipulate the equation for the desired expression.

Rearrange the equation to isolate the desired expression: \(a^2 + b^2 = 34ab\) Divide both sides by \(ab\): \(\frac{a}{b} + \frac{b}{a} = 34\) Therefore, the correct answer is (4) 34.

Key concepts

Understanding Logarithmic Equations

Logarithmic equations involve the logarithm function, which is essentially the inverse of exponentiation. The property that a logarithm of a product equals the sum of the logarithms (\( \text{log}(xy) = \text{log}(x) + \text{log}(y) \) ) and that a logarithm of a power allows the exponent to be brought in front of the log (\( \text{log}(x^n) = n\text{log}(x) \) ) are particularly useful when solving equations that contain logarithms.

For example, to solve the given equation \( \text{log} \left(\frac{a+b}{6}\right) = \frac{1}{2}(\text{log} a + \text{log} b) \) from our exercise, we use these properties to combine logarithmic terms and eliminate the logarithmic function. This is a critical step in finding the solution as it simplifies the problem into a more recognizable algebraic form.

Exponents and Radicals in Algebra

Exponents and radicals are important in the world of algebra. They allow us to express repeated multiplication concisely and to represent roots of numbers. In our exercise, we encounter a square root (also written as a fractional exponent such as \( a^{\frac{1}{2}} \) ), which arises from squaring both sides to eliminate the square root.

Why Square Both Sides?

By squaring both sides of the equation \( a + b = 6(a^{\frac{1}{2}}*b^{\frac{1}{2}}) \) we're using the property that \( (x^{\frac{1}{2}})^2 = x \) which means the square of a square root returns the original number. This method is a common technique to isolate variables that are under a radical.

Algebraic Manipulation to Isolate Variables

Effective algebraic manipulation is essential for solving complex equations. It involves rearranging and simplifying equations using basic operations and properties of equality to isolate the desired variables.

In the final steps of our solution, algebraic manipulation comes into play by rearranging the equation \( a^2 + 2ab + b^2 = 36ab \) to isolate the expression \( \frac{a}{b} + \frac{b}{a} \). This involves identifying that \( a^2 + b^2 \) and \( 2ab \) are parts of the expanded form of \( (a + b)^2 \) and recognizing that we can manipulate the equation to cast it in a form where \( ab \) is a common factor, allowing us to divide through and reach our desired outcome.