Question

Decide whether each equation is true or false. Justify your answer. Assume \(c, x,\) and \(y\) are positive real numbers and \(c \neq 1\). a) \(\frac{\log _{e} x}{\log _{e} y}=\log _{e} x-\log _{e} y\) b) \(\log _{c}(x+y)=\log _{c} x+\log _{c} y\) c) \(\log _{c} c^{n}=n\) d) \(\left(\log _{c} x\right)^{n}=n \log _{c} x\) e) \(-\log _{c}\left(\frac{1}{x}\right)=\log _{c} x\)

Short answer

a) False, b) False, c) True, d) False, e) True

Step-by-step solution

- Analyze equation (a)

Check if \(\frac{\frac{\text{log}\thinspace x}{\text{log}\thinspace y}}{\text{log}\thinspace y} = \text{log} \thinspace x - \text{log}\thinspace y\) is true or false. The LHS \(\frac{\text{log}\thinspace x}{\text{log}\thinspace y}\) implies the ratio of two logarithms, whereas the RHS expresses the difference of logarithms. These expressions are not equivalent in logarithmic rules. Therefore, equation (a) is false.

- Analyze equation (b)

The equation \(\text{log}_c(x + y) = \text{log}_c x + \text{log}_c y\) suggests a rule similar to the product rule of logarithms. However, logarithmic properties state that \(\text{log}_c(xy) = \text{log}_c x + \text{log}_c y\), not for sums. Thus, equation (b) is false.

- Analyze equation (c)

The equation \(\text{log}_c c^n = n\) follows directly from the definition of logarithms: \(\text{log}_c c^n\) means 'to what power must c be raised, for it to become \(c^n\)?' The answer is \(n\). So, equation (c) is true.

- Analyze equation (d)

For \((\text{log}_c x)^n = n \text{log}_c x\), notice that the LHS implies exponentiation of the logarithm, whereas the RHS implies multiplication of the logarithm by \(n\). These are fundamentally different operations involving logarithms. Therefore, equation (d) is false.

- Analyze equation (e)

The equation \(-\text{log}_c(\frac{1}{x}) = \text{log}_c x\) is equivalent to \(\text{log}_c(x^{-1})\) using logarithm properties: \(\text{log}_c(\frac{1}{x}) = \text{log}_c(x^{-1}) = -\text{log}_c x\). Therefore, equation (e) is true.

Key concepts

logarithmic properties

Logarithms come with a set of properties that help simplify complex expressions and solve equations more easily. Here are a few essential logarithmic properties that you need to know:

- **Product Rule**: States that the logarithm of a product is the sum of the logarithms of the factors. Mathematically, \(\begin{align*} \text{log}_c(xy) = \text{log}_c x + \text{log}_c y \).
- **Quotient Rule**: Expresses that the logarithm of a quotient is the difference of the logarithms. This is written as \(\begin{align*} \text{log}_c \frac{x}{y} = \text{log}_c x - \text{log}_c y \).
- **Power Rule**: Indicates that the logarithm of a number raised to a power is the power multiplied by the logarithm of the number. This can be shown as \(\begin{align*} \text{log}_c(x^n) = n \text{log}_c x \).

Each of these properties plays a crucial role in simplifying logarithmic expressions. Correctly applying these rules will allow for a clearer understanding and justification of why certain logarithmic equations hold true or false.

exponential functions

Exponential functions are an important type of function in mathematics, frequently seen in precalculus and higher-level math courses. An exponential function can be defined as follows:

\(\begin{align*} f(x) = a \times b^x \)

Here, \(a\) is a constant, \(b\) is the base of the exponential function, and \(x\) is the exponent. The properties of exponential functions are essential for dealing with logarithms because logarithms are inherently the inverse of exponentials.

One of the core concepts understood is the inverse relationship between logarithms and exponentials: for any positive real number \(x\) and any positive base \(c eq 1\),:\(\begin{align*} \text{log}_c(x) = y \ \text{if and only if} \ c^y = x \)

This inverse relationship helps in transforming between logarithmic and exponential forms, crucial for solving many types of mathematical problems. The exponential function's inherent growth properties make it handy in various fields, from natural sciences to finance.

logarithmic identities

Logarithmic identities are special equations that are always true and can be used to simplify logarithmic expressions. Here are some important logarithmic identities:

- **Change of Base Formula**: This vital identity allows you to evaluate logarithms with bases other than 10 or \(e\). It's expressed as \( \begin{align*} \text{log}_a b = \frac{\text{log}_c b}{\text{log}_c a} \).
- **Logarithm of 1**: Any logarithm of 1 to any base is always 0. Formally, \( \begin{align*} \text{log}_c 1 = 0 \).
- **Logarithm of the Base**: This identifies that the logarithm of the base itself is always 1: \( \begin{align*} \text{log}_c c = 1 \).

Using these identities, you can reframe and simplify some logarithmic expressions, which is crucial when trying to determine the equality or prove certain logarithmic relationships. This knowledge allows us to verify or disprove the equations as shown in your precalculus problems.

For example, \( -\text{log}_c \frac{1}{x} = \text{log}_c x \) and \( \text{log}_c c^n = n \) are identities that help in verifying equations.