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. \(2=10^{\log _{10} 2^{2}}\) 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

Question: Determine if the following statements are true or false. Explain your reasoning or provide a 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) \(2=10^{\log _{10} 2^{2}}\) 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 \(f^{-1}(x)=1 / x\)

Step-by-step solution

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

The original equation is \(y=3^{x}\), where y is explicitly expressed as an exponential function of x. To find the inverse, we need to get x in terms of y. Taking the natural log (base-e) of both sides, we get: \(\ln y = \ln(3^{x})\) Using the logarithm property, multiply x with ln(3): \(\ln y = x\cdot \ln(3)\) Now, divide both sides by \(\ln(3)\) to get x in terms of y: \(x = \frac{\ln y}{\ln(3)}\) This result is different and not equal to \(\sqrt[3]{y}\). Therefore, the statement is false. A counterexample: if y = 27, then x = \(\frac{\ln 27}{\ln 3}\) = 3, but \(\sqrt[3]{27} = 3\).

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

This statement is false because the correct logarithm property is called quotient rule, which states that: \(\frac{\log _{b} x}{\log _{b} y}=\log _{b}\left(\frac{x}{y}\right)\) A counterexample: let b = 10, x = 100, and y = 10, then \(\frac{\log _{10} 100}{\log _{10} 10} = \frac{2}{1} = 2 \neq \log _{10}(100)-\log _{10}(10) = 1\)

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

This statement is false, and we can prove it by simplifying both sides. Use the logarithm property called "power rule": \(\log _{5} 4^{6}=6\cdot \log _{5} 4\) A counterexample: let b = 5, x = 4, and y = 6, then \(6\cdot \log _{5} 4 \neq 4\cdot \log _{5} 6\)

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

Use the logarithm power rule, and rewrite 2 as 10^1: \(10^{\log _{10} 2^{2}} = 10^{\log _{10} 4}\) Since the base and the base of the logarithm are the same, they cancel out, and we get: 4 The statement is false. A counterexample: the statement claims that 2=10^{\log _{10} 2^{2}}}, but we've shown that 4=10^{\log _{10} 2^{2}}}.

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

Use the logarithm power rule, and rewrite 2 as \(\ln e^1\): \(\ln 2^{e} = e\cdot \ln 2\) The statement is false. A counterexample: the statement claims that \(2=\ln 2^{e}\) but we've shown that \(e\cdot \ln 2=\ln 2^{e}\).

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

To find the inverse of the function, replace f(x) with y and solve for x: \(y=x^{2}+1\) Subtract 1 from both sides: \(y-1 = x^2\) Now, take the square root of both sides (considering only the principal square root for simplicity) to get x in terms of y: \(x=\sqrt{y-1}\) This result is different and not equal to \(1 /\left(x^{2}+1\right)\). Therefore, the statement is false. A counterexample: if x = 1, then f(1) = 2, but \(f^{-1}(2)=1 /\left(2^{2}+1\right)=\frac{1}{5}\); however, we found that \(f^{-1}(2)=\sqrt{2-1}=1\).

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

To find the inverse of the function, replace f(x) with y and solve for x: \(y=1 / x\) Multiply both sides by x and y: \(xy=1\) Now, divide both sides by y to get x in terms of y: \(x=\frac{1}{y}\) Indeed, \(f^{-1}(x)=\frac{1}{x}\), so the statement is true.

Key concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They are often written in the form of f(x) = bx, where b is the base and x is the exponent. These functions grow at a rate proportional to their value, which is why they are used to model phenomena with rapid growth, such as population increase or radioactive decay.

In the context of statement 'a' from the original exercise, y = 3x is an exponential function with a base of 3. It shows how y changes in response to different values of x. The solution diligently explains that the inverse function is not the cube root of y but rather x = ln(y) / ln(3), demonstrating the exponential function's characteristic of increasing geometrically; doubling x more than doubles y.

Inverse Functions

Inverse functions effectively reverse the effect of the original function. Given a function f(x), its inverse, denoted as f-1(x), satisfies the relationship f(f-1(x)) = x. It reverts the output back to the original input. For example, if we have a function that squares a number, its inverse would extract the square root, assuming the inputs are non-negative for the sake of simplicity.

When working with inverse functions, such as in statements 'f' and 'g', it's important to switch the roles of x and y and then solve for the new x. A common mistake is to assume operations such as squaring or taking reciprocals will be simply inverted without due process. The solutions clearly illustrate how to algebraically derive the correct inverse operations.

Counterexample Methodology

The counterexample methodology is a logical approach to disprove a statement by finding a single instance where the statement does not hold true. This is a powerful strategy in mathematics, as it takes only one counterexample to prove that a universal statement is false. When dealing with abstract concepts, such as the properties of logarithms or functions, counterexamples provide concrete evidence of the fallacies.

In the provided solutions, the counterexample methodology is used effectively to demonstrate the falsity of certain statements. For example, in statements 'a', 'b', and 'c', specific numbers are substituted into the equations, showing that the purported equality does not hold, thereby negating the original claims. This emphasizes the value of counterexamples in constructing rigorous mathematical arguments.

Natural Logarithm

The natural logarithm, denoted as ln(x), is a special logarithm with a base of e, where e (approximately 2.71828) is an irrational and transcendental number known as Euler's number. Natural logarithms are commonly used in calculus, complex analysis, and scientific fields for their special properties related to growth processes, and because they simplify certain types of differential and integral equations.

In statements 'a' and 'e', the natural logarithm is involved. In these cases, the specific properties of the natural logarithm are employed to solve for variables and to demonstrate that certain statements are false when assessed through these properties. As the logarithmic counterpart to exponential functions, ln(x) undoes the effects of raising e to a power, a principle that is subtly illustrated in the solutions of these statements.