Question

Determine whether the following statements are true and give an explanation or counterexample. a. If \(y=3^{x}\), then \(x=\sqrt[3]{y}\). b. \(\frac{\log _{b} x}{\log _{b} y}=\log _{b} x-\log _{b} y\) c. \(\log _{5} 4^{6}=4 \log _{5} 6\) d. \(2=10^{\log _{10} 2}\) e. \(2=\ln 2^{e}\) f. If \(f(x)=x^{2}+1,\) then \(f^{-1}(x)=1 /\left(x^{2}+1\right)\). g. If \(f(x)=1 / x,\) then \(f^{-1}(x)=1 / x\).

Short answer

In summary: - Statement a is true: \(x=\sqrt[3]{y}\) is equivalent to \(x=\log_{3}{y}\) - Statement b is false: \(\frac{\log _{b} x}{\log _{b} y}\) is not equal to \(\log _{b} x-\log _{b} y\) - Statement c is false: \(\log _{5} 4^{6}\) is not equal to \(4 \log _{5} 6\) - Statement d is true: \(2=10^{\log _{10} 2}\) - Statement e is false: \(2\) is not equal to \(\ln 2^{e}\) - Statement f is false: \(f^{-1}(x)=1 /\left(x^{2}+1\right)\) is not the correct inverse function - Statement g is true: the inverse function of \(f(x)=1 / x\) is indeed \(f^{-1}(x)=1 / x\)

Step-by-step solution

Statement a: \(y=3^{x}\) then \(x=\sqrt[3]{y}\)

Since \(y=3^{x}\), we can write this as \(x=\log_{3}{y}\). This statement is true, as it is the inverse function of the given one, and they are equivalent.

Statement b: \(\frac{\log _{b} x}{\log _{b} y}=\log _{b} x-\log _{b} y\)

This statement is false. We know that \(\frac{\log _{b} x}{\log _{b} y} = \log_b{\left(\frac{x}{y}\right)}\). The given statement would be true if it was written as \(\log _{b} x - \log _{b} y= \log_b{(\frac{x}{y})}\).

Statement c: \(\log _{5} 4^{6}=4 \log _{5} 6\)

This statement is false. Using the rules of logarithms, \(\log _{5}4^{6} = 6\log_{5}{4}\). So the statement should be written as \(\log _{5} 4^{6}= 6 \log _{5} 4\).

Statement d: \(2=10^{\log _{10} 2}\)

This statement is true. By definition, \(b^{\log_{b}{x}} = x\), so in this case, \(2=10^{\log_{10}{2}}\).

Statement e: \(2=\ln 2^{e}\)

This statement is false. Using the logarithmic identity, \(\ln{2^e} = e\ln{2}\). Since \(e\) is the base of the natural logarithm, the statement should be written as \(e=\ln e^{2}\).

Statement f: \(f(x)=x^{2}+1,\) then \(f^{-1}(x)=1 /\left(x^{2}+1\right)\)

Let's find the inverse function of \(f(x)=x^2+1\): 1. Replace \(f(x)\) with \(y\): \(y = x^2 + 1\) 2. Swap \(x\) and \(y\): \(x = y^2 + 1\) 3. Solve for \(y\): \(y = \pm\sqrt{x - 1}\) This statement is false. The inverse function should be written as \(f^{-1}(x)=\pm\sqrt{x - 1}\).

Statement g: If \(f(x)=1 / x,\) then \(f^{-1}(x)=1 / x\)

To find the inverse function of \(f(x) = \frac{1}{x}\): 1. Replace \(f(x)\) with \(y\): \(y = \frac{1}{x}\) 2. Swap \(x\) and \(y\): \(x = \frac{1}{y}\) 3. Solve for \(y\): \(y = \frac{1}{x}\) This statement is true, as the inverse function of \(f(x)=\frac{1}{x}\) is indeed \(f^{-1}(x)=\frac{1}{x}\).

Key concepts

logarithmic identities

Logarithmic identities are essential in understanding how logarithms behave under various algebraic operations. A basic identity is the product rule, which states that

• \( \log_b{(xy)} = \log_b{x} + \log_b{y} \)

This identity says that the logarithm of a product is the sum of the logarithms. Another important identity is the quotient rule:

• \( \log_b{(\frac{x}{y})} = \log_b{x} - \log_b{y} \)

This rule indicates that the logarithm of a quotient is the difference of the logarithms. The power rule is also crucial:

• \( \log_b{(x^r)} = r\log_b{x} \)

This tells us that the logarithm of a number raised to a power is the power times the logarithm of the number. These identities are useful for simplifying logarithmic expressions or solving logarithmic equations.

exponential functions

Exponential functions are mathematical functions of the form \( f(x) = a^x \), where \( a \) is a positive constant, and \( a eq 1 \). These functions grow or decay at constant percent rates. A key feature of exponential functions is their constant rate of change, which is not linear but rather multiplicative. The graph of an exponential function is always a curve that either grows rapidly as \( x \) increases if the base is greater than 1, or decreases if \( 0 < a < 1 \).

A property of exponential functions is that the inverse of an exponential function is a logarithmic function. If you have an exponential equation like \( y = a^x \), then its inverse is \( x = \log_a{y} \). This characteristic underpins many applications in science, finance, and other fields where exponential growth or decay is present.

function inverses

Function inverses are essential when you need to reverse the effect of a function. For a function \( f(x) \), its inverse, \( f^{-1}(x) \), will satisfy the property:

• \( f(f^{-1}(x)) = x \)
• \( f^{-1}(f(x)) = x \)

This means if you begin with a number, apply the function, and then apply the inverse function, you'll arrive back at your original number.
It's crucial for a function to be one-to-one to have an inverse. This means for every input, there is a unique output and vice versa. Methods for finding an inverse involve:

• Replacing \( f(x) \) with \( y \)
• Swapping \( x \) and \( y \)
• Solving for \( y \)

The ability to find inverse functions has practical use in solving equations, including those involving exponentials and logarithms, where you often switch between these forms to isolate variables and find solutions.